aa r X i v : . [ m a t h . AG ] F e b A NOTE ON 3-SUBGROUPS IN THE SPACE CREMONA GROUP
KONSTANTIN LOGINOV
Abstract.
We prove that a finite 3-group in the Cremona group Cr ( C ) can be generated by atmost 4 elements. This provides the last missing piece in bounding the ranks of finite p -subgroupsin the space Cremona group. Introduction
We work over the field C of complex numbers. The Cremona group Cr n ( C ) is the group ofbirational self-maps of the n -dimensional complex projective space P n . For n = 1, this group isisomorphic to PGL(2 , C ). In contrast, for n ≥ n = 2 by I. Dolgachev and V. Iskovskikh, see [DI09]. In dimension 3,the situation is more delicate, and the complete classification seems to be out of reach. However,there exist classificational results for some classes of groups, see e.g. [Pr09] in the case of simplegroups. There are also numerous boundedness results for finite subgroups of the Cremona group,see [PS16] and references therein.We concentrate on one particular type of such subgroups, namely, the p -groups. By a p -group we mean a finite group of order p k where p is a prime number. A rank r ( G ) of a p -group G isdefined as the minimal number of elements that generate G . The following problem is natural:estimate the rank of a p -subgroup in Cremona group Cr n ( C ), cf. [Ser09]. The complete answer tothis question for n = 2 was obtained by A. Beauville, see Theorem 1.4.In higher dimensions, one can consider the group of birational self-maps of a rationally connectedvariety of a given dimension, and ask for a similar bound for the rank of its p -subgroups. For n = 3,the sharp bounds were obtained for p = 2 and p ≥
5, see Theorem 1.3. For n = 3, p = 3 in thework [Kuz20] the bound r ( G ) ≤ Theorem 1.1.
Let X be a -dimensional rationally connected variety, and let G ⊂ Bir( X ) be afinite -group. Then r ( G ) ≤ , and this bound is sharp. Corollary 1.2.
Let G ⊂ Cr ( C ) be a finite -group. Then r ( G ) ≤ , and this bound is sharp. As a consequence, we can formulate the following theorem which provides sharp bounds for theranks of p -subgroups acting on rationally connected threefolds. Theorem 1.3 (cf. [Pr11], [Pr14], [PS17], [Xu20]) . Let X be a rationally connected variety ofdimension , and let G ⊂ Bir( X ) be a finite p -group. Then it can be generated by at most r elements where • r = 6 when p = 2 , • r = 4 when p = 3 , • r = 3 when p ≥ .Moreover, these bounds are sharp. In the assumptions of Theorem 1.3, from [Xu20] it follows that for p ≥ G is abelian.For the discussion of analogous statements in the case p = 2 , p = 2, Example 2.6 for p = 3and the natural action of ( Z /p ) on P for p ≥
5. Let us formulate an analogous statement indimension 2.
Theorem 1.4 ([Beau07]) . Let X be a rational surface, and let G ⊂ Bir( X ) be a finite abelian p -group. Then it can be generated by at most r elements where • r = 4 when p = 2 , r = 3 when p = 3 , • r = 2 when p ≥ .Moreover, these bounds are sharp. The assumption that G is abelian can be removed from Theorem 1.4 due to Proposition 2.9. Infact, Beauville proves that if the bounds as above are attained then one can describe the conjugacyclass of the given subgroup. It would be interesting to obtain analogous description in the 3-dimensional case, cf. Question 1.9 in [PS17]. Interesting results in this direction were obtained in[Xu18].We give a sketch of the proof of Theorem 1.1. We assume that a 3-group G of rank ≥ G is abelian: this can be achieved by passing to the quotient group by its Frattini subgroup,see Proposition 2.9. First we use a standard technique: resolve indeterminacy of the action of G on X and apply a G -MMP to obtain a G Q -Mori fiber space with a faithful action of G . Then,using known results on 3-groups in the plane Cremona group, we exclude the case of a positive-dimensional base. So we have to deal only with the case of a 3-dimensional G Q -Fano variety. Inthis case, we consider the action of G on the anti-canonical linear system which turns out to benon-empty by orbifold Riemann-Roch theorem, see Section 4. More precisely, there are two casesto examine: either the anti-canonical system contains one element up to multiplication by a scalar,or its dimension is at least 2.In the first case, we study the geometry of the anti-canonical divisor, see Section 5. We show thatit is a reduced, irreducible surface that admits an action of a 3-group of rank r ( G ) −
1. Moreover,it is equivariantly birational either to a K3 surface, or to a rational surface, or to a ruled surfaceover an elliptic curve. In all these subcases we use estimates on the rank of 3-groups that acts onsuch surfaces (see Section 3) to derive a contradiction.In Section 6 we deal with the case when the anti-canonical system has dimension ≥
2. Since G is abelian, we can find a G -invariant pencil of anti-canonical elements. If the singularities of thispencil are “bad enough”, in Proposition 6.2 we prove that our variety is G -birational to a G Q -Morifiber space with the base of positive dimension, and we are done. Otherwise, we obtain a pencil ofK3 surfaces with at worst canonical singularities. Again, using estimates on the rank of 3-groupsacting on such surfaces, we arrive at a contradiction.Our approach differs from that of [Kuz20]: we do not use the classificational results on three-dimensional Gorenstein Fano varieties. This may be useful for possible generalisations. Acknowledgements.
The author thanks Yuri Prokhorov for reading the draft of the paper andmany valuable comments, Alexandra Kuznetsova and Constantin Shramov for useful discussions.The author is a Young Russian Mathematics award winner and would like to thank its sponsorsand jury. 2.
Preliminaries
We work over the field of complex numbers. All the varieties are projective and defined over C unless stated otherwise. We use the language of the minimal model program (the MMP for short),see e.g. [KM98], and the MMP for linear systems, cf. [Al94].2.1. Contractions.
By a contraction we mean a projective morphism f : X → Y of normalvarieties such that f ∗ O X = O Y . In particular, f is surjective and has connected fibers. A fibration is defined as a contraction f : X → Y such that dim Y < dim X .Let G be a finite group. By a G -variety we mean a variety X together with an action of a group G . If f is a G -equivariant contraction (resp., a G -equivariant fibration) of G -varieties, we call it a G -contraction (resp., a G -fibration ).2.2. Pairs and singularities. A pair ( X, B ) consists of a normal variety X and a boundary Q -divisor B with coefficients in [0 ,
1] such that K X + B is Q -Cartier. A pair ( X, B ) over Z is apair ( X, B ) together with a projective morphism X → Z . Let φ : W → X be a log resolution of( X, B ) and let K W + B W = φ ∗ ( K X + B ). The log discrepancy of a prime divisor D on W withrespect to ( X, B ) is 1 − µ D B W and it is denoted by a ( D, X, B ). We say (
X, B ) is lc (resp. klt)(resp. ǫ -lc) if a ( D, X, B ) is ≥ > ≥ ǫ ) for every D . Note that if ( X, B ) is ǫ -lc,then automatically ǫ ≤ a ( D, X, B ) = 1 for almost all D . lc-place of ( X, B ) is a prime divisor D over X , that is, on birational models of X , such that a ( D, X, B ) = 0. A lc-center is the image on X of a lc-place.We say that a group G acts on the pair ( X, B ) if X is a G -variety and the boundary divisor D is G -invariant.2.3. Log Fano and log Calabi-Yau pairs.
Let (
X, B ) be an lc pair over Z . We say ( X, B ) is log Fano over Z if − ( K X + B ) is ample over Z . If Z is a point then ( X, B ) is called a log Fanopair . In this case if B = 0 then X is called a Fano variety . We say (
X, B ) is log Calabi-Yau over Z if K X + B ∼ Q Z . If Z is a point then ( X, B ) is called a log Calabi-Yau pair .2.4.
Mori fiber space. A G Q - Mori fiber space is a G Q -factorial variety X with at worst terminalsingularities together with a G -contraction f : X → Z to a variety Z such that ρ G ( X/Z ) = 1 and − K X is ample over Z . If X is G -factorial (e.g. if X is smooth), we call it G - Mori fiber space .We formulate results of O. Haution that are applicable to the study of p -subgroups in theCremona group. Recall that a Chern number of a smooth projective variety is the intersectionnumber of Chern classes of its tangent bundle.
Theorem 2.1 ([Ha19, 1.1.2]) . Let G be a p -group, and let X be a smooth projective variety endowedwith an action of G . Assume that either G is abelian, or dim X < p . If G has no fixed points on X , then every Chern number of X is divisible by p . Corollary 2.2.
Let p > be a prime number, and let G be a abelian p -group. Suppose that X is a terminal G Q -factorial threefold whose non-Gorenstein singularities are of the type (1 , , .Moreover, assume that G has no fixed points on X . Then the integer ( − K X ) is divisible by p .Proof. Since non-Gorenstein singular points on X are of type (1 , , − K X ) = a where a ∈ Z . Let f : X ′ → X be a G -equivariant resolution [AW97]. We may assume that f is anisomorphism outside finite number of singular points of X . Write K X ′ = f ∗ K X + X a i E i . Since 2 is invertible modulo p , we can consider the following formula modulo p :( − K X ′ ) = (cid:16) − f ∗ K X − X a i E i (cid:17) = ( − K X ) − X Q a Q where a Q is a contribution made by exceptional divisors over a singular point Q ∈ X . By Theorem2.1, the integer ( − K X ′ ) is divisible by p . Since G has no fixed points on X , the exceptionaldivisors E i form G -orbits whose length is divisible by p , so it follows that P Q a Q is equal to 0modulo p . Note that 2 ( − K X ) is an integer since − K X is a Cartier divisor. Hence 2 ( − K X ) is divisible by p , and the claim follows. (cid:3) Theorem 2.3 ([Ha19, 1.2.1]) . Let G be a p -group, and let X be a smooth projective variety withan action of G . Assume that either G is cyclic, or dim X < p − . Then G has no fixed points on X if and only if the Euler characteristic χ ( X, F ) of every G -equivariant coherent O X -module F isdivisible by p . Corollary 2.4 ([Xu20, Main theorem]) . Let X be a rationally connected variety of dimension n.If G ⊂ Bir( X ) is a p -subgroup and p > n + 1 , then G is abelian and the rank of G is at most n . As a consequence, if n = 3 then any p -group G ⊂ Bir( X ) for p ≥ ≤ p -group. Example 2.5.
Consider a faithful action of the group ( Z / on P given by the matrices (cid:18) (cid:19) , (cid:18) − (cid:19) , Then the group G = ( Z / faithfully acts on P × P × P . This shows that the bound r ( G ) ≤ p = 2. Example 2.6.
Let S be a Fermat cubic surface defined by the equation x + x + x + x = 0 n P . Let the group ( Z / act on S by multiplication by a cube root of unity on of the coordinates x i , 1 ≤ i ≤
3. Then the group G = ( Z / × Z / X = S × P where the action ofa second factor Z / G of rank 4 canbe embedded in Cr ( C ), so the bound r ( G ) ≤ p = 3. Example 2.7.
For p = 3 there exists an action of a non-abelian group on a rational threefold.Indeed, consider an action of the group G = H × Z / Z of rank 3 on X = P × P . Here H = ( Z / × Z / ⋊ Z / Question 2.8.
Assume that for p = 2 (resp., p = 3) a p -group G with r ( G ) = 6 (resp., r ( G ) = 4)faithfully acts on a rationally connected 3-dimensional variety. Is it true that G is abelian?The next proposition shows that to bound the rank of p -groups acting on a rationally connectedvariety it is enough to bound the rank of abelian p -groups. Proposition 2.9 ([Kuz20, 3.1]) . Assume that for all rationally connected varieties Y of dimension n and any abelian p -subgroup A ⊂ Bir( Y ) we have r ( A ) ≤ N for some number N . If X is rationallyconnected and of dimension n , then for any p -subgroup G ⊂ Bir( X ) (in particular, non-abelian)we have r ( G ) ≤ N . We include the idea of its proof for the reader’s convenience. Recall that a
Frattini subgroup Φ( G ) of a p -group G is an intersection of all maximal subgroup of G . It is normal and has thefollowing remarkable property: the quotient group G/ Φ( G ) is isomorphic to an elementary abelian p -group ( Z /p ) r ( G ) . Passing to a resolution of indeterminacy of the group action, we may assumethat a p -group G faithfully acts on a rationally connected variety X of dimension n . Then thequotient group G/ Φ( G ) of rank r ( G ) faithfully acts on a rationally connected variety X/ Φ( G ), andthe claim follows.The next proposition bounds the rank of a 3-group that acts on a three-dimensional terminalsingularity. Proposition 2.10 ([Pr11, 2.4], [Kuz20, 4.4]) . Assume that X is a threefold with terminal singu-larities and G is a -subgroup in Aut( X ) fixing a point x ∈ X . Then r ( G ) ≤ . G -action on K3 surfaces In this section, we consider actions of p -groups on (possibly, singular) K3 surfaces and theirgeneralisations. By a K3 surface we mean a normal surface S with at worst canonical (that is,1-lc) singularities such that H ( O S ) = 0 and K S ∼ Proposition 3.1.
For p ≥ , assume that a p -group G faithfully acts on a K3 surface S . Then r ( G ) ≤ .Proof. Passing to the minimal resolution we may assume that S is a smooth K3 surface with afaithful action of the group G . Then the claim follows from the work of V. Nikulin [Nik80]. Weprovide some details. There exists an exact sequence1 → Aut s ( S ) → Aut( S ) → Z /I → s ( S ) is the subgroup of symplectic automorphisms, that is, automorphisms that preservea holomorphic 2-horm on S , and I is some integer. Put G s = G ∩ Aut s ( S ). Then | G s | ≤ r ( G s ) ≤
1. But since the image of G in Z /I is cyclic, the group G has rank ≤ (cid:3) Now we consider actions of p -groups on log Calabi-Yau surface pairs, see 2.3 for the definition.Such pairs are generalize the notion of K3 surfaces. Let ( S, ∆) be a lc log Calabi-Yau surface pairendowed with an action of a p -group G . In particular, the surface S is normal and the boundary Q -divisor ∆ is G -invariant. Proposition 3.2.
For p ≥ , assume that a p -group acts on a log Calabi-Yau surface pair ( S, ∆) which is lc but not -lc (in particular, this is satisfied either if ∆ = 0 or ∆ = 0 and S is notcanonical). Then r ( G ) ≤ . roof. Using the assumption that ( S, ∆) is not 1-lc and taking the minimal resolution of S , wemay run the G -MMP to obtain a G -Mori fiber space S ′ → Z where S ′ is a smooth surface, and Z is either a point or a curve.If Z is a point then S ′ is isomorphic to P , but a p -group of rank > P faithfully,for example, by Theorem 1.4. Hence we may assume that Z is a smooth curve. It is well knownthat in this case Z is either a rational curve or a curve of genus 1, see e.g. [BL20, 4.1]. If Z is arational curve then S ′ is rational, so again r ( G ) ≤ Z is a smooth curve of genus 1. Note that Z admits a faithful action ofa p -group G ′ of rank r ( G ) −
1. It is well known that, after fixing the base point on Z , we haveAut( Z ) ≃ Z ⋊ Z /k where Z is identified with the subgroup of translations, and k can be equalto 2 , ,
6. For p ≥
5, clearly G ⊂ Z , and hence G ′ sits in the p -torsion subgroup of Z whichis isomorphic to ( Z /p ) . Therefore in this case r ( G ′ ) ≤ r ( G ) ≤
3. For p = 3, we haveto pay attention to the case k = 3 and k = 6. In this case, there is a subgroup isomorphic to( Z / × Z / ⋊ Z / Z ). However, one checks that the rank of such subgroup is equal to 2,hence r ( G ′ ) ≤
2. Therefore r ( G ) = 3, and the proposition is proven. (cid:3) Anticanonical linear system
In this section, we estimate the dimension of the anti-canonical linear system H ( X, O X ( − K X ))of a G Q -Fano variety X that admits an action of an abelian 3-group of rank ≥
5. The assumptionthat G is abelian is justified by Proposition 2.9.4.1. Orbifold Riemann-Roch.
By [Re85, 10.2], for a terminal threefold X and a Weil Q -Cartierdivisor D on it we have the following version of the Riemann-Roch formula:(4.1) χ ( O X ( D )) = χ ( O X ) + 112 D ( D − K X )(2 D − K X ) + 112 D · c ( X ) + X Q c Q ( D )where for any cyclic quotient singularity Q we have(4.2) c Q ( D ) = − i r − r + i − X j =1 bj ( r − bj )2 r . Here r is the index of Q , the divisor D has type i r ( a, − a,
1) at Q , b satisfies ab = 1 mod r , anddenotes the residue modulo r . Non-cyclic non-Gorenstein singularities correspond to a basket ofcyclic points. Moreover, by [Re85, 10.3] one has(4.3) ( − K X ) · c ( X ) + X Q ( r − /r ) = 24The following proposition shows that the number of non-Gorenstein points is bounded by 9under our assumptions. We omit its proof which follows from Proposition 2.10 and formulas (4.1),(4.3). Proposition 4.4 ([Kuz20, 4.9]) . Assume that a -group G of rank r acts on a G Q -factorialterminal Fano threefold X , and that X is non-Gorenstein. Then r ≤ and if r = 5 then non-Gorenstein points on X have type (1 , , and form one orbit of length . From now on we assume that r ( G ) = 5. In the next proposition we compute the dimension ofthe anti-canonical system on X under this assumption. Proposition 4.5.
Let X be a G Q -Fano threefold where G is a -group of rank . Then either • ( − K X ) = 1 / and dim | − K X | = 0 , or • dim | − K X | ≥ .Proof. Consider two cases: when X is Gorenstein and when it is not. In the first case, we mayuse the usual Riemann-Roch formula (4.1) without the correction terms c Q . From Kawamata-Viehweg vanishing theorem it follows that χ ( O X ( − K X )) = h ( O X ( − K X )) and χ ( O X ) = 1, so for D = − K X we obtain 4 ≤ g + 2 = h ( O X ( − K X )) = 3 + 12 ( − K X ) , where the number g is called the genus of X . The claim is proven in this case. ow assume that X is non-Gorenstein. For D = − K X and singular points of type (1 , ,
1) wehave c Q = − /
8, so if there are N such points, in the formula (4.1) they give a contribution − N/ − K X ) · c ( X ) = 24 − N . We have(4.7) h ( O X ( − K X )) = 3 + 12 ( − K X ) − N N = 9, using ( − K X ) ≥ / h ( O X ( − K X )) = 34 + 12 ( − K X ) ≥ h ( O X ( − K X )) = 1 if and only if ( − K X ) = 1 /
2, and if ( − K X ) ≥ h ( O X ( − K X )) ≥ (cid:3) G -action on threefolds. Special case In this section, we work in the following setting.
Setting 5.1.
Let X be a terminal G Q -factorial Fano threefold where G is an abelian 3-group ofrank r = 5. Assume that ( − K X ) = 1 / | − K X | = 0. Denote the unique anti-canonicalelement in | − K X | by S .This case can be excluded by Corollary 2.2. However, we give an alternative proof. Proposition 5.2 ([Pr11, 2.9]) . The pair ( X, S ) is lc.Sketch of proof. If the pair (
X, S ) is not lc then pick a maximal λ <
X, λS )is lc. Note that it is an lc log Fano pair. Consider its minimal lc-center Z . After perturbing thepair by a G -invariant divisor with small coefficients we may assume that Z is a unique, and hence G -invariant, lc-center of the lc log Fano pair ( X, ∆). Moreover, Z is either a point or a smoothrational curve. Considering an action of G on Z and on the normal bundle to the generic point on Z and using Proposition 2.10, we conclude that in both cases we have r ( G ) ≤ (cid:3) Proposition 5.3. S is irreducible and reduced.Proof. By our assumptions S = 1 / S is Cartier. Assume that S is either reducible ornon-reduced. Write S = P k i S i in Cl( X ) for some k i ≥
1. The intersection number (2 S ) · D fora Cartier divisor 2 S and an effective Weil divisor D is an integer, see e.g. [KM98, 1.34]. We have(2 S ) · D > X k i S i · (2 S ) ≥ X k i and conclude that either S = S + S or S = 2 S . Since G is a 3-group, in both cases thecomponents of S are G -invariant.Since ρ G ( X ) = 1, it follows that the components of S are proportional to S in Cl( X ) modulothe torsion subgroup. By Proposition 5.2 we may assume that the pair ( X, S ) is lc. Then thenormalization S ν of S is a lc del Pezzo surface endowed with an action of a 3-group G ′ of rank r ( G ) − G on S and on the tangentspace to its generic point. If S ν is klt then it is rational and we derive a contradiction to Theorem1.4. If S ν is strictly lc then we may apply the same argument as in Proposition 5.2 for λ = 1 toderive a contradiction. This shows that S is irreducible and reduced. (cid:3) Proposition 5.4.
The case r ( G ) = 5 as in the setting 5.1 does not occur.Proof. By Proposition 5.3 we may assume that S is reduced and irreducible. Considering theaction of G on the tangent space to the generic point of S , we see that a p -group G ′ of rank 4faithfully acts on S . Let ν : S ′ → S be the normalization of S . Consider the pair ( S ′ , ∆ ′ ) definedby the formula K S ′ + ∆ ′ = ν ∗ K S ∼ . Here ∆ ′ is an effective Weil Q -divisor called the different , see [Ka07]. Moreover, ν (∆ ′ ) is supportedon the non-normal locus of S , the pair ( S ′ , ∆ ′ ) is lc, and ∆ ′ is G ′ -invariant. If ∆ ′ = 0 and S ′ = S s canonical, we apply Proposition 3.1. Indeed, we have K S ∼ → O X ( K X ) → O X → O S → h ( O S ) = 0, so S is a K3 surface with at worst canonical singularities. By Proposition3.1 we conclude that r ( G ′ ) ≤
2, hence r ( G ) ≤
3. If either ∆ ′ = 0 or ∆ ′ = 0 and S ′ = S is notcanonical then we apply Proposition 3.2 and obtain r ( G ′ ) ≤
3, so r ( G ) ≤
4. This contradicts tothe assumption r ( G ) = 5, and the proposition if proven. (cid:3) G -action on threefolds. General case In this section, we work in the following setting. Let X be a terminal G Q -factorial Fano threefoldwhere G is a 3-group of rank r ( G ) = 5, and ( − K X ) > / G is abelian. Then by Proposition 4.5 we havedim | − K X | ≥
1, and since a representation of G in H ( X, O ( − K X )) splits a direct sum of 1-dimensional representations, we see that |− K X | has a G -invariant linear subsystem H of dimension2. We will need the following simple lemma. Lemma 6.1.
Assume that a p -group of rank ( Z /p ) r for p > faithfully acts on a -dimensionalrationally connected G Q -Mori fiber space X → Z with the base of positive dimension. Then r ( G ) ≤ δ p, .Proof. Consider an exact sequence 1 → G → G → G → G acts faithfully on the base Z , and G acts faithfully acts on the generic fiber of X → Z .If dim Z = 1 then Z ≃ P , and G ⊂ Z /p , so G acts faithfully acts on a smooth geometricallyrational surface and hence has rank ≤ δ p, according to Theorem 1.4, so the claim follows. Ifdim Z = 2, we argue analogously. (cid:3) The next proposition is the crucial technical tool in this paper.
Proposition 6.2 (cf. [Al94, 3.1], [Pr09, 6.6]) . Assume that X is a normal terminal threefoldendowed with a faithful action of a finite group G , and let H be a non-empty G -invariant pencil(that is, a linear system of dimension ) on X without fixed components, such that − K X − H ∼ E where E is an effective Q -divisor. Assume that either • E = 0 , and the pair ( X, H ) is not canonical, or • E = 0 ,Then X is G -birationally equivalent to a G Q -Mori fiber space with the base of positive dimension.Proof. We start with the first case. Assume that the linear system H does not have fixed compo-nents, and that the pair ( X, H ) is not canonical. Let c be its canonical threshold, that is, c is maxi-mal with the property that ( X, c H ) is canonical. Clearly, 0 < c <
1. Let f : ( X ′ , c H ′ ) → ( X, c H )be a G -equivariant G Q -factorial terminalization. We have K X ′ + c H ′ = f ∗ ( K X + c H ) ,K X ′ + H ′ + E = f ∗ ( K X + H ) ∼ , where E is a non-zero effective integral (since H ∼ − K X ) f -exceptional Q -divisor. Note that E may be reducible. Since − K X − c H is ample, we have that − K X ′ − c H ′ is big and nef. We runa G -equivariant ( K X ′ + c H ′ )-MMP and obtain a G Q -Mori fiber space ( X, H ) → Z .Note that X is terminal. Indeed, we show that each step of ( X ′ , c H ′ )-MMP is also a step ofthe usual K X ′ -MMP. This follows from the fact that on the terminalization X ′ the linear system H does not contain curves in its base locus [Al94, 1.22], and hence it is nef. On each step, the pairremains terminal, so its base locus does not contain curves, and hence the linear system remainsnef, so the same argument applies on each step of the MMP.We may assume that Z is a point, otherwise the proposition is proven. So ρ G ( X ) = 1, and wehave K X + H + E ∼ , rom [Al94, 3.1] it follows that E = 0. Then − K − H ∼ E, and we have reduced the first case of the lemma to the second.Now we consider the second case, so we assume that E = 0. Pass to a G Q -factorial terminal-ization f : ( X ′ , H ′ ) → ( X, H ). Write K X ′ + H ′ + E ′ = f ∗ ( K X + H + E ) ∼ E ′ = 0. Apply a G -MMP to the pair ( X ′ , H ′ ). On each step of this MMP we contract a( K X ′ + H ′ )-negative extremal ray. But − ( K X ′ + H ′ ) ∼ E ′ , hence E ′ cannot be contracted bythe negativity lemma. Hence the G -MMP terminates with a G Q -Mori fiber space X → Z . If thedimension of Z is positive, the proposition is proven. So we assume that Z is a point, ρ G ( X ) = 1,and K X + H + E ∼ . Write − K X ∼ a H for some a >
1. As explained above, the pair ( X, H ) is terminal. Hence by[Al94, 1.22] the general member H ∈ H is smooth and contained in the smooth locus of X . Byadjunction formula we obtain K H = ( K X + H ) | H = ( a − H | H , hence H is a smooth del Pezzo surface. Note that the linear system H defines a rational map X P . Resolve its base locus of H and run a relative G -MMP over P to obtain a Mori fiberspace with the base of positive dimension. (cid:3) Corollary 6.3.
Let X be a -dimensional G Q -Fano variety that admits a G -invariant anti-canonical pencil, that is, a G -invariant linear system H such that dim H = 1 , and H ⊂ | − K X | .Then either • X is G -birationally equivalent to a G Q -Mori fiber space with the base of positive dimension,or • the pair ( X, H ) is canonical, the linear system H does not have fixed components, and X is G -birationally equivalent to a G -equivariant fibration over P whose general fiber isa smooth K3 surface.Proof. By Proposition 6.2, we may assume that the pair ( X, H ) is canonical and does not havefixed components.We resolve indeterminacy of the rational map X P given by H . Consider a terminalization( X ′ , H ′ ) → ( X, H ). Then the linear system H ′ is basepoint free.We have K X ′ + H ′ = f ∗ ( K X + H ) ∼ , and the pair ( X ′ , H ′ ) is terminal. From [Al94, 1.23] it follows that H ′ may have only isolatedbasepoints (that are non-singular on X ′ ) such that the multiplicity of a g.e. in H ′ at them equals1. We claim that H ′ cannot have isolated basepoints. Indeed, since H ′ is generated by twoelements that are strict preimages of D , D on X , the base locus of H ′ coincides with theirintersection. But since the base locus lives in the non-singular locus of X ′ , two surfaces cannotintersect in a point there, so the base locus if H ′ is empty. Hence the general element of H ′ is asmooth K3 surface, and the corollary is proven. (cid:3) Hence we obtain a G -morphism X ′ → P . But then a group of rank r ( G ) − Proof of the main results
Proof of Theorem 1.1.
Assume that G is a 3-group of rank ≥ Y . By Proposition 2.9 we may assume that G is abelian. Passingto a G -equivariant resolution of singularities and running a G -MMP, we may assume that G faithfully acts on a G Q -Mori fiber space X → Z . By Lemma 6.1, we deduce that Z is a point, so X is a 3-dimensional G Q -factorial Fano variety with ρ G ( X ) = 1.By Proposition 4.5 either ( − K X ) = 1 / | − K X | = 0, or dim | − K X | ≥
1. We showthat the first case does not occur. For a unique element S ∈ | − K X | , by Proposition 5.2 we ay assume that the pair ( X, S ) is lc. Then, by Proposition 5.3 the divisor S is reduced andirreducible. Proposition 5.4 shows that S is a K3 surface with at worst canonical singularities, butthen a 3-group of rank r ( G ) − ≥ S by Proposition 3.2.So we see that dim | − K X | ≥
1. Hence there exists a G -invariant pencil H , that is, a G -invariant subsystem of dimension 1, of anti-canonical elements. We consider the pair ( X, H ).Applying Proposition 6.2 and Lemma 6.1 we deduce that H does not have base components andthat the pair ( X, H ) is canonical. But then by Corollary 6.3 we have that X is G -birational toa fibration over P whose general fiber is a smooth K3 surface. A group of rank r ( G ) − r ( G ) ≤
4. The fact that this bound is sharp follows from Example 2.6. (cid:3)
Proof of Corollary 1.2.
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