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Mori dream spaces of Calabi-Yau type and the log canonicity of the Cox rings

We prove that a Mori dream space over a field of characteristic zero is of Calabi-Yau type if and only if its Cox ring has at worst log canonical singularities. By slightly modifying the arguments we also reprove the characterization of the varieties of Fano type by the log terminality of the Cox rings.


Introduction
A normal projective variety X is said to be of Fano type if there exists a boundary Q-divisor ∆ on X such that (X, ∆) has klt singularities and −(K X + ∆) is ample. [BCHM10] proved that a Q-factorial variety of Fano type is a Mori dream space. Then [GOST12, Theorem 1.1] proved that a Q-factorial variety of Fano type can be characterized as a Mori dream space whose Cox ring has at worst log terminal singularities. In [GOST12] they also considered the Calabi-Yau version and proved a similar result by assuming some conjecture (see [GOST12,Theorem 4.13]). The conjecture has been partially verified in [FT11,Theorem 3.3], so that their statement is generalized in the corresponding cases [GOST12,Theorem 4.10].
In this paper we give a conjecture-free proof to the Calabi-Yau version of the result. Namely, we verify the following theorem.
Theorem 1.1. Let X be a Mori dream space over a field of characteristic zero. Then X is of Calabi-Yau type if and only if the Cox ring of X has at worst log canonical singularities.
A normal projective variety X is said to be of Calabi-Yau type if (X, ∆) has log canonical singularities and K X +∆ is numerically trivial for some boundary Q-divisor ∆. Note that a variety of Calabi-Yau type is not necessarily a Mori dream space.
By slightly modifying the arguments, we can also reprove [GOST12, Theorem 1.1] in Theorem 4.1. During the preparation of this paper, we received a preprint from Morgan Brown [B11] in which he independently gave a different proof to the 'only if' directions of Theorem 1.1 and Theorem 4.1.
We explain a little bit about the proof of Theorem 1.1. For the 'only if' direction, we first take a set of r = rank Pic (X) ample divisors which are linearly independent in Pic (X) Q , and take the affine space bundle π : Y = Spec X Sym( i O X (A i )) → X. From a boundary Q-divisor ∆ on X which makes the pair (X, ∆) log Calabi-Yau, we construct a boundary divisor ∆ Y on Y such that the pair (Y, ∆ Y ) is also log canonical and K Y + ∆ Y is Q-linearly trivial. By contracting the zero section of π, we obtain a birational morphism f : Y → Z such that (Z, ∆ Z ) is log canonical and K Z + ∆ Z is Q-linearly trivial for some Qdivisor ∆ Z . It turns out that there exists a small birational morphism from the spectrum of the Cox ring of X to Z. Thus we prove the log canonicity of the Cox ring.
In our proof of the 'if' direction of Theorem 1.1, we first derive the Q-effectivity of the anti-canonical divisor of X from the log canonicity of the Cox ring, by using a similar construction as above and applying the numerical characterization of the pseudo-effective divisors due to [BDPP04] (for a Mori dream space, this result is much easier to prove; see Proposition 2.6). Since X is assumed to be a Mori dream space, we can apply the anti-canonical MMP which terminates in a semi-ample model in the same way as in [GOST12]. By the standard facts from [HK00], the semi-ample model is the quotient of an open subset of the spectrum of the Cox ring by the dual torus action of Pic (X). We can assume that the action is free, so that the log canonicity descends to the quotient. Therefore the semi-ample model is of Calabi-Yau type. As demonstrated in [GOST12, Proof of Theorem 1.2], we can trace back the anti-canonical MMP to prove that X itself is of Calabi-Yau type.
Throughout the paper, we work over a field k of characteristic zero.
The first author was supported by Grant-in-Aid for Scientific Research (A) 22244002. The second author was supported by Grant-in-Aid for JSPS fellows 22-849, and by the GCOE program "Research and Training Center for New Development in Mathematics".

Preliminaries
We start with the definitions of varieties of Fano type and Calabi-Yau type.
Definition 2.1 (cf. [PS09, Lemma-Definition 2.6]). Let X be a projective normal variety over a field and ∆ an effective Q-divisor on X such that K X + ∆ is Q-Cartier.
(i) We say that (X, ∆) is a klt Fano pair if −(K X + ∆) is ample and (X, ∆) is klt. We say that X is of Fano type if there exists an effective Q-divisor ∆ on X such that (X, ∆) is a klt Fano pair. (ii) We say that X is of Calabi-Yau type if there exits an effective Q-divisor ∆ such that K X + ∆ ∼ Q 0 and (X, ∆) is log canonical. Next we give the definition of Mori dream spaces.
Definition 2.2 (cf. [HK00]). A normal projective variety X over a field is called a Mori dream space if X satisfies the following three conditions: (i) X is Q-factorial, Pic (X) is finitely generated, and Pic (X) Q ≃ N 1 (X) Q , (ii) The nef cone Nef(X) is the affine hull of finitely many semi-ample line bundles, (iii) there exists a finite collection of small birational maps f i : X X i such that each X i satisfies (i) and (ii), and that the closed movable cone Mov(X) is the union of the f * i (Nef(X i )). The following is one of the most important properties of Mori dream spaces. The notion of multi-section rings and the Cox rings are repeatedly used in the studies of Mori dream spaces.
Definition 2.4 (Multi-section rings and Cox rings). Let X be an integral normal scheme. For a semi-group Γ of Weil divisors on X, the Γ-graded ring is called the multi-section ring of Γ. Suppose that the divisor class group Cl (X) is finitely generated. For such X, choose a group Γ of Weil divisors on X such that Γ Q → Cl (X) Q is an isomorphism. Then the multi-section ring R(X, Γ) is called a Cox ring of X.
Remark 2.5 (See [GOST12, Remark 2.18] for details). As seen above, the definition of Cox rings depends on the choice of the group Γ. We can prove that the ambiguity does not affect the basic properties of rings, such as finite generation, log terminality (log canonicity), etc. In fact, there is a canonical way to define Cox rings without ambiguity (up to isomorphisms) due to Hausen [Hau08]. We can check that the properties of rings mentioned above holds for his Cox ring if and only if ours has the same properties. Since our definition of Cox rings as multi-section rings is easier for calculation, we adopt our definition in this paper.
We need the following intersection-theoretic characterization of effective (resp. big) divisors on a Mori dream space.
Proposition 2.6. Let D be a divisor on a Mori dream space X. Then X is Q-effective (resp. big) if and only if its intersection number with any curve in any covering family is at least (strictly greater than) zero.
Proof. Note that the Q-effectivity and the pseudo-effectivity are equivalent for divisors on a Mori dream space. It is enough to show the following statement.
Claim. Let X be a Mori dream space and D a not big but Q-effective (resp. not Q-effective) divisor on X. Then there exists a curve in a covering family C on X such that C · D = 0 (resp. C · D < 0).
Proof. We run a D-MMP so that the pushforward D N of D on X N is semi-ample (resp. there exists a D N -Mori fiber space from X N ). Note that all the birational maps above are surjective in codimension one. Therefore we can find an open subset U of X N whose complement has codimension at least two such that all the birational maps above are identities on U.
If D is not big (resp. not Q-effective), we take D N -Iitaka fiber space (resp. D N -Mori fiber space) f : X N → Y . Since the relative dimension of f is positive, there is a curve C ′ in a covering family which is contained in a fiber of f . Since X N \ U has codimension at least two, we can choose C ′ so that it is contained in U. Now let C be the strict transform of C ′ on X. We see that this C has the desired properties.  Proof. Choose ample line bundles A 1 , . . . , A r on X which are linearly independent in Pic (X) R , where r = rank Pic (X). Consider the following natural morphism where Γ ′ is the semi-group generated by the classes of A i , and let π : Y → X to be the structure morphism. Since A i are ample, f is a birational projective morphism which contracts the zero section of π, which we denote again by X (see [Har66,Proposition 3.5]).
If (X, ∆) is a log canonical pair, then it follows that Y, concluding that (Z, ∆ Z ) is a log canonical pair.
We shall derive the log canonicity of the Cox rings of X.

Claim. The corresponding morphism
is birational and contracts no divisor.
Proof. Choose an ample divisor A 0 from the interior of the cone spanned by Γ ′ . Then for any positive integer m > 0 and a non-zero global section s ∈ H 0 (X, O X (mA 0 )), the natural ring homomorphism By [HK00, Lemma 2.7], if we take two global sections s 1 and s 2 of mA 0 such that the corresponding divisors on X have no common component, then {s 1 , s 2 } ⊂ R(X, Γ) is a regular sequence. Therefore the divisors of s 1 and s 2 have no common irreducible components, and this concludes the proof.
From the claim above, we see that the pair (Spec R(X, Γ), ∆ ′ ) is also log canonical, where ∆ ′ is the effective Q-divisor naturally defined from ∆ Z . Since Spec R(X, Γ) is Q-factorial by [HK00, Proposition 2.9], we see that Spec R(X, Γ) itself is log canonical.
3.2. Mori dream spaces with log canonical Cox rings. In this subsection we prove the 'if' direction of Theorem 1.1.
Theorem 3.2. Let X be a Mori dream space whose Cox rings have at worst log canonical singularities. Then X is of Calabi-Yau type. Proof.
Step 1. We prove that −K X is Q-effective.
Let C be an arbitrary irreducible curve on X which belongs to a covering family. We take a set of divisors D 1 , . . . , D r on X with the following properties.
• They forms a basis of Cl (X) Q , so that if we set Γ = r i=1 ZD i , R(X, Γ) is a Cox ring of X.
• The effective cone of X is contained in the cone spanned by D 1 , . . . , D r , so that holds.
• C · D i ≥ 0 holds for i = 1, . . . , r. We note that the choice of the D i depends on C. Let Y be the total space of the direct sum i O X (−D i ): and regard X ⊂ Y as the zero section. Then we have H 0 (Y, O Y ) ∼ = R(X, Γ). Therefore we have a natural birational morphism f : Y → Z = Spec R(X, Γ) which is not necessarily proper. Proof. If the assertion does not hold, then there exists a curve C ′ on Y which is not contained in the union of the E i and mapped to a point by f . Since the morphism f is defined by the elements of R(X, Γ), for any s ∈ R(X, Γ), the value of s is constant on C ′ . We note that C ′ is not necessarily complete.
Take a positive linear combination D = d i D i which is very ample. For a global section 0 = s ∈ H 0 (X, D), we see that div Y (s) = π * div X (s)+ d i E i , where π : Y → X is the structure morphism. Suppose that π(C ′ ) is not a point. Then we can take a section s ∈ H 0 (X, D) such that the divisor div Y (s) does not contain C ′ but intersect C ′ . Hence s is not constant on C ′ , a contradiction. If π(C ′ ) is a point, C ′ is a curve in the fiber π −1 (π(C ′ )), which is identified with A r by choosing a non-zero local section of O X (D i ) around π(C ′ ) for each i. If π(C ′ ) is not contained in div X (s), we see that the restriction of s to the fiber is a non-zero monomial of exponent (d 1 , . . . , d r ) under the identification. Since C ′ is assumed to be not contained in the coordinate hyperplanes, we can find a suitable exponent (d 1 , . . . , d r ) such that the monomial is not constant on C ′ .
Since Z has only log canonical singularities, we can write It follows that (K X · C) ≤ 0. Since C was arbitrary, we conclude that −K X is Q-effective by Proposition 2.6.
Step 2. Since X is a Mori dream space, we have a (−K X )-MMP where each step is a birational map and −K X N is semi-ample, because −K X is Q-effective.
Fix a Cox ring R(X, Γ) of X. We recall some facts on the GIT of Cox rings from [HK00, Proposition 2.9]. First of all, there exists a canonical action of the torus T = Hom(Γ, k * ) on the affine variety Spec R(X, Γ), and X N is the categorical quotient by T of the semistable locus U ⊂ Spec R(X, Γ) with respect to a character of T which corresponds to an ample divisor on X N ([HK00, Proof of Proposition 2.9]).
Since X is Q-factorial and X N is obtained from a MMP starting from X, we see that X N also is Q-factorial. Q-factoriality of X N , in turn, implies that the quotient U → U/T = X N is the geometric quotient (see [HK00, Proposition 1.11(2)] and [HK00, Corollary 2.4]). Moreover, by replacing Γ with its subgroup of finite index if necessary, we can assume that the torus T acts on U freely ([HK00, Proposition 2.9]). Therefore the quotient morphism U → U/T = X N is smooth.
Since Spec R(X, Γ) is log canonical, so is its open subset U. Hence we see that X N = U/T has log canonical singularities. By taking a general effective Q-divisor ∆ N on X N which is Q-linearly equivalent to −K X N , we obtain a log Calabi-Yau pair (X N , ∆ N ) due to the semiampleness of −K X N .
Finally we trace back the (−K X )-MMP as in [GOST12, Proof of Theorem 1.2], showing that X itself is of Calabi-Yau type.

Characterization of varieties of Fano type revisited
Using the similar arguments as above, we can reprove the characterization of varieties of Fano type (=[GOST12, Theorem 1.1]).
Theorem 4.1. Let X be a Q-factorial normal projective variety over k. X is of Fano type if and only if the Cox ring of X is of finite type and log terminal.
Proof. For 'only if' direction it is enough to prove that the Cox ring of X is log terminal, since the finite generation is proved in [BCHM10, Corollary 1.3.2]. We only point out which part of the proof of Theorem 3.1 should be modified.
Instead of thinking of the pair Y, we should think of the following klt pair for some positive number ǫ < 1. Then it holds that (K Y + ∆ Y )| X = K X + ∆ + ǫ i A i , since we have O Y (E i )| X ≃ O X (−A i ). By replacing A i from the beginning if necessary, we can assume that there exists ǫ such that −(K X + ∆) ∼ Q ǫ i A i , so that (K Y + ∆ Y )| X ∼ Q 0.
Therefore we get the equality as before, where ∆ Z = f * ∆ Y . Hence the pair (Z, ∆ Z ) is klt. The rest of the proof is the same. For the proof of the 'if' direction, we first prove that −K X is big. If we carry out the same arguments as in Step 1 of the proof of Theorem 3.2, by the log terminality of the Cox ring we see that e i < 1 holds for all i. Since some positive linear combination of the D i is ample, we deduce that C · (−K X ) = C · ( (1 − e i )D i ) > 0. Since X is a Mori dream space, this implies the bigness of −K X by Proposition 2.6.
Next, by arguing as in Step 2, we arrive at a model X N such that −K X N is semi-ample and big. We can also show that X N is log terminal from the log terminality of the Cox ring, concluding that X N is of Fano type. The rest is the same.