aa r X i v : . [ m a t h . AG ] N ov WEAKLY-EXCEPTIONAL QUOTIENT SINGULARITIES
DMITRIJS SAKOVICS
Abstract.
A singularity is said to be weakly-exceptional if it hasa unique purely log terminal blow up. In dimension 2, V. Shokurovproved that weakly-exceptional quotient singularities are exactlythose of types D n , E , E , E . This paper classifies the weakly-exceptional quotient singularities in dimensions 3 and 4. Contents
1. Introduction 12. Preliminaries 53. Three-dimensional case 124. Four-dimensional case 14References 231.
Introduction
Let G ⊂ GL N ( C ) be a finite subgroup. Then G has a natural linearaction on C N . The aim of this paper is to study the exceptionalityproperties of quotient singularities C N /G in low dimensions.The Chevalley-Shephard-Todd theorem (see [23, Theorem 4.2.5])easily implies that as far as the exceptionality of the induced quotientsingularities is concerned, one can assume that G contains no quasi-reflections. Furthermore, the exceptionality of the singularity is onlydependent of the group’s induced action on P N − (for details, see Sec-tion 2.3), so one can assume that G is actually a subgroup of SL N ( C ).Let ¯ G be the image of G in the natural projection of SL N ( C ) ontoPGL N ( C ). Since the two groups are closely related, the argument willmove from considering ¯ G to considering G and back at will.Standard notation will be used throughout this paper, but for theconvenience of the reader and in order to prevent any ambiguity, thenotation used will be fully described in Section 2.1.Since the singularities are defined by group actions, one needs todistinguish between different types of actions of subgroups of GL N ( C ). Note that any G ⊂ GL N ( C ) comes equipped with an action on C N .Hence the properties of the action of G can be said to be properties of G (or ¯ G ) itself. Definition 1.1.
The group G is irreducible if for any non-zero vector x ∈ C N , the G -orbit of x spans C N . Definition 1.2.
The group G is primitive if there does not exist adecomposition of C N into a direct sum of proper linear subspaces, suchthat G permutes the subspaces. If such a decomposition does exist, thenthe action is imprimitive . Definition 1.3.
An imprimitive group G is monomial , if the action of G is induced from a -dimensional action of some subgroup G ′ ⊆ G .In other words, G = D ⋊ T , where D consists of diagonal matrices,and T ⊆ S N acts by permuting the basis. In order to avoid repetitions,throughout this paper “monomial” will be used to mean “irreduciblemonomial” (in particular, T is assumed to be transitive). From the more geometric point of view, it is more interesting to lookat the variety C n /G . Note that since this is a quotient singularity, itis a Kawamata log terminal singularity. Definition 1.4.
Let ( V ∋ O ) be a germ of a Kawamata log terminalsingularity. The singularity is said to be exceptional if for every ef-fective Q -divisor D V on the variety V , such that the log pair ( V, D V ) is log canonical, there exists at most one exceptional divisor over thepoint O with discrepancy − with respect to the pair ( V, D V ) . Proposition 1.5 (see [20, Proposition 2.1]) . Let G ⊂ SL N ( C ) be a fi-nite subgroup that induces an exceptional singularity. Then G is prim-itive. Corollary 1.6.
For any given N , only finitely many finite subgroupsof SL N ( C ) induce exceptional singularities.Proof: Immediate by Proposition 1.5 and Jordan’s theorem (see, forexample, [11]). (cid:3)
One can define a larger class of singularities by looking at a specialclass of birational morphisms:
Theorem 1.7 (see [6, Theorem 3.7]) . Let ( V ∋ O ) be a germ ofa Kawamata log terminal singularity. Then there exists a birationalmorphism π : W → V such that the following hypotheses are satisfied: • the exceptional locus of π consists of one irreducible divisor E such that O ∈ π ( E ) , EAKLY-EXCEPTIONAL QUOTIENT SINGULARITIES 3 • the log pair ( W, E ) has purely log terminal singularities. • the divisor − E is a π -ample Q -Cartier divisor.Proof: The result follows from [1], [16, Theorem 1.5] and [19, Propo-sition 2.9] (cid:3)
Definition 1.8.
Let ( V ∋ O ) be a germ of a Kawamata log terminalsingularity, and π : W → V be a birational morphism satisfying theconditions of Theorem 1.7. Then π is a plt blow-up of the singularity. Definition 1.9.
We say that the singularity ( V ∋ O ) is weakly-exceptional if it has a unique plt blow-up. Lemma 1.10 (see [19, Theorem 4.9]) . If ( V ∋ O ) is exceptional, then ( V ∋ O ) is weakly-exceptional. Theorem 1.11.
Let G ⊂ SL N ( C ) be a finite subgroup that induces aweakly-exceptional singularity. Then G is irreducible.Proof: The argument is similar to that in [20, Proposition 2.1]. (cid:3)
This result will be used frequently throughout this paper, since forany finite group G , it allows to severely limit the number of actionsworth checking. Example 1.12.
One can consider the case N = 2 . It is a well-knownfact that the finite subgroups of Aut ( P ) are (up to conjugation): cyclic( Z n , n ≥ ), dihedral ( D n , n ≥ ) or polyhedral ( A , S , A ), with liftsto SL ( C ) being their central extensions by Z (except the case of oddcyclic groups, where the central extension is not necessary). Cyclicgroups induce reducible actions, polyhedral groups induce primitive ac-tions, and the dihedral groups induce monomial actions.On the geometric side, rephrasing [22, Section 5.2.3] implies that A , S and A give rise to exceptional singularities of types E , E and E respectively, the dihedral groups D n give rise to weakly-exceptional, butnot exceptional singularities of type D n +2 and the cyclic groups Z n giverise to singularities of type A n − , that are not weakly-exceptional. Theconcept of a weakly-exceptional singularity was originally formed as ageneralisation of the singularities of types D n and E n . Since this example fully settles the N = 2 case, assume for the restof this paper that N ≥
3. In the dimensions N ≤
6, the exceptionalsingularities have been fully classified in [6], [7] and [17]. The aim of thispaper is to obtain a list of group actions that induce weakly-exceptionalsingularities in dimension 3 and 4. This will be done by applyingthe exceptionality criteria obtained by Y. Prokhorov, I. Cheltsov andC. Shramov in [6] and [20], using the language of Tian’s alpha-invariantintroduced in [24] and [25] by G. Tian and S-T. Yau.
DMITRIJS SAKOVICS
For dimension N = 3, the classification of finite subgroups of SL ( C )is a classical result by G. Miller, H. Blichfeldt and L. Dickson (see [18]).A modern exposition of this result has been made by S. Yau and Y. Yu(see [26, Theorem A]). A result by D. Markushevich and Y. Prokhorovsays: Theorem 1.13 (see [17]) . The group G induces an exceptional singu-larity if and only if ¯ G is isomorphic to A , Klein’s simple group K of size , Hessian group H of size or its normal subgroup F of size . The first main result of this paper extends this by:
Theorem 1.14.
Let G ⊂ SL ( C ) be a finite subgroup. Then G inducesa weakly-exceptional but not an exceptional singularity if and only if oneof the following holds: • G is a monomial group, and ¯ G is not isomorphic to ( Z ) ⋊ Z or ( Z ) ⋊ S . • G is isomorphic to the normal subgroup E ⊳ F of size . Section 3 of this paper is devoted to proving this theorem.Since 4 is not a prime number, SL ( C ) contains significantly morefinite subgroups than SL ( C ) does. The list of finite primitive sub-groups of SL ( C ) is a classical result, that can be found in H. Blich-feldt’s book (see [2, Chapter VII]). The list of irreducible imprimitivefinite subgroups can be obtained from papers by D. Flannery (see [10])and B. H¨ofling (see [12]).The second main result of this paper is designed to supplement thelist of finite subgroups of SL ( C ) inducing exceptional singularities,that can be found in a paper by I. Cheltsov and C. Shramov (see [6]).In order to avoid reproducing the tables of groups from the works men-tioned above, the paper will instead produce a list of irreducible groupactions that give rise to singularities that are not weakly-exceptional.Therefore, the second main result of this paper is: Theorem 1.15. If G ⊂ SL ( C ) is a finite subgroup whose action in-duces a singularity that is not weakly-exceptional, then either G is notirreducible, or ¯ G must be conjugate to one of (using the notation de-scribed in Section 2.2 where appropriate): • A primitive group that is one of: – S , A . – ( H , H , H , H ) ∼ = H × H , for different choicesof H , H ∈ { A , S , A } . – ( A × A ) ⋊ Z , ( S × S ) ⋊ Z , ( A × A ) ⋊ Z . EAKLY-EXCEPTIONAL QUOTIENT SINGULARITIES 5 – [ S × S ] ∼ = ( A × A ) ⋊ Z ∼ = ( S , A , S , A ) . – (( A × A ) ⋊ Z ) ⋊ Z ∼ = ( S , A , S , A ) ⋊ Z . • An imprimitive non-monomial group that is one of: – D m × H , where H ∈ { A , S , A } . – [ D m × S ] ∼ = ( Z m × A ) ⋊ Z ∼ = ( D m , Z m , S , A ) . – [ D m × S ] ∼ = ( D m × A ) ⋊ Z ∼ = ( D m , D m , S , A ) for m ≥ . – [ D m × S ] ∼ = ( Z m × V ) ⋊ S ∼ = ( D m , Z m , S , V ) . • A monomial group that is one of: – (cid:0) ( Z ) ⋊ Z (cid:1) ⋊ Z , acting as shown in Lemma 4.2. – ( Z ) ⋊ S , acting as shown in Lemma 4.2. – A or S , acting as twisted diagonal groups ( H , , H , α . – D m × D n ( m, n ≥ ). – ( D n × D n ) ⋊ Z ( m, n ≥ ). – D n ⋊ Z ( n ≥ ). – A ⋊ Z , S ⋊ Z . – [ S × S ] ∼ = ( V × V ) ⋊ S ∼ = ( S , V , S , V ) . – [ A × A ] ∼ = ( V × V ) ⋊ Z ∼ = ( A , V , A , V ) . – [ D m × D n ] ∼ = ( Z m × D n ) ⋊ Z ∼ = ( D m , Z m , D n , D n ) ( m, n ≥ ). – [ D m × D n ] α ∼ = ( Z m × Z n ) ⋊ D ∼ = ( D m , Z m , D n , Z n ) α (where α ( b ) = a n n , α ( a m m ) = b ). – [ D m × D n ] ∼ = ( D m × D n ) ⋊ Z ∼ = ( D m , D m , D n , D n ) ( m, n ≥ ). – (( V × V ) ⋊ S ) ⋊ Z ∼ = ( S , V , S , V ) ⋊ Z . – (( V × V ) ⋊ Z ) ⋊ Z ∼ = ( A , V , A , V ) ⋊ Z . – (( Z m × Z m ) ⋊ D ) ⋊Z ∼ = ( D m , Z m , D m , Z m ) α ⋊Z (where α ( b ) = a m m , α ( a m m ) = b ). – (( D m × D m ) ⋊ Z ) ⋊ Z ∼ = ( D m , D m , D m , D m ) ⋊ Z for m ≥ . Section 4 of this paper is devoted to proving this result.
Remark 1.16.
Note that the list above gives explicit conjugacy classesof most of the subgroups. In most cases, the irreducible subgroup isuniquely defined (up to conjugacy) by its isomorphism class.
The results in this paper can be applied in birational geometry, in-cluding the problem of conjugacy of subgroups of higher-dimensionalCremona groups. For more details, see [4].2.
Preliminaries
DMITRIJS SAKOVICS
Notation.
In this section, some standard notation used through-out this paper is defined: • Z n is the cyclic group of size n . • D n is the dihedral group of size 2 n . • S n is the permutation group of a set of n elements. • A n is the alternating group, an index 2 subgroup of S n . • Z n , D n , A n , S n ⊂ SL ( C ) are the binary versions of the relevantgroups, i.e. their central extensions by Z (see, for example,[23, Section 4.4]). Since the generators of these groups will bereferred to heavily at the end of Section 4, fix the presentationsof these groups as: – Z n = – D n = – A = < [12][34] , [14][23] , [123] > , where the basis is chosen sothat [12][34] preserves the two lines spanned by the basisvectors, and [14][23] swaps them. Extend this presentationto that of S = < [12][34] , [14][23] , [123] , [34] > . – A = < [12345] , [12][34] > , where the basis is chosen so that[12345] is diagonal.The last 3 groups are central extensions of permutation groups,and their generators are intentionally named to identify thepermutations they correspond to. The relations come from therelations between the permutations. • V is the Klein group of size 4. • ζ n is a primitive n -th root of unity. Whenever two such are usedin defining generators of the same group, the choice is assumedto be consistent, i.e. ζ aab = ζ b for a, b ∈ Z . • Write “extension of G by scalar elements” (for a finite group G ) to mean H ⋊ G ⊂ SL N ( C ), where H is a subgroup of thecentre of SL N ( C ) (i.e. consists of scalar matrices). Note thatunless N is prime, there are several non-trivial possibilities for H that give H ⋊ G the same image under the natural projectionto PGL N ( C ).Note that the groups V , D and Z × Z are isomorphic. The notationused for this group will denote the context the group is considered in: V will be used whenever it is considered as a subgroup of A or S ,and one of the other two will be used whenever it is considered on itsown.2.2. Group actions on a smooth quadric surface in P . In thissection some notation for a specific type of group action on P will bepresented. This is a general form for an action that preserves a smooth EAKLY-EXCEPTIONAL QUOTIENT SINGULARITIES 7 quadric surface in P (which can be taken to be P × P embeddedvia the Segre embedding). Some of the notation was taken from thisaction’s description in [9, Section 4.3], with some additions tailored todescribing individual members of families of related groups.It is possible to present P as the set of non-zero 2 × P to be: ( x : y : u : v ) (cid:18) x yu v (cid:19) Let S be the image of P × P under the Segre embedding into P .Then one can assume S is the zero set of the determinant of the matrixform of P .Let ¯ G be a finite group acting faithfully on P × P . This varietyhas exactly two rulings, which ¯ G can either preserve or interchange.Consider the exact sequence1 −→ H −→ ¯ G −→ S where S permutes the two rulings. Then H E ¯ G is the maximal sub-group that preserves the ruling, and either ¯ G = H or ¯ G is generatedby H and an element στ , where τ is the involution interchanging thetwo rulings, and σ is some automorphism of S preserving the ruling,with σ ∈ H (see [9, Theorem 4.9]).Let π i : H → H i be the projections of H on the two components of P × P . Then have two more short exact sequences1 −→ K −→ H −→ H −→ −→ K −→ H −→ H −→ K ∩ K = { } . Therefore, for i = j ( i, j ∈ { , } ), K i ∼ = ˆ K i := { kK j | k ∈ K i } E H/K j = H i In this notation, H / ˆ K ∼ = H/ ( K K ) ∼ = H / ˆ K , so the group can bedefined completely by ( H , K , H , K ) α , where α is an isomorphism H / ˆ K → H / ˆ K . In return, if H is known, one can reconstruct α bymaking it map π ( h ) ˆ K π ( h ) ˆ K ( ∀ h ∈ H ).In the matrix form described above, H acts on P by left matrixmultiplication, and H acts by transposed right matrix multiplication.The involution switching the two rulings of P × P corresponds totransposing the matrix. If h ∈ H acts on the first component of P × P and h ∈ H acts on the second one, then write h h | h i to DMITRIJS SAKOVICS denote this action. Explicitly, for any 2 × A and B , h A | B i (cid:18)(cid:18) x yu v (cid:19)(cid:19) = A (cid:18) x yu v (cid:19) B T Furthermore, interchanging the rulings of P × P corresponds to trans-posing the matrix form of P . It is worth noting that if ¯ G = H , thenconjugation by στ provides isomorphisms H ∼ = H and K ∼ = K .The following notation may sound somewhat unnecessary, but it willassist in avoiding describing explicit generators of groups later on. Forsome groups, one can say some of its elements are of some special“type”, e.g. the order 2 element in D n (n odd) or the 3-cycles in A .Given a finite subgroup ( H , K , H , K ) α with H , H ⊂ SL ( C ), sayelements of H and H of some fixed type are “coupled”, if ∀ h h | h ′ i ∈ H with h an element of this type, then h ′ must be either an element of thesame type or a product of such an element and an element of a differenttype. Otherwise say that elements of this type are “not coupled”. Forexample, if H ∼ = H ∼ = D , then order 2 elements b are not coupled in G if G contains an element (cid:10) a k | ba l (cid:11) for some k, l ∈ Z , where a is anorder 5 element. Otherwise they are coupled.2.3. Exceptionality criteria.
In order to determine the exceptional-ity of a given singularity, it is more useful to have a more computablecriterion than that given by the definitions of exceptionality and weakexceptionality. To discuss this criterion, some definitions need to bemade.
Definition 2.1.
Let X be a smooth Fano variety (see [14] ) of dimen-sion n , and let g = ( g ij ) be a K¨ahler metric, such that ω = √− π X g ij d z i ∧ d ¯ z j ∈ c ( X ) Let ¯ G ⊆ Aut ( X ) be a compact subgroup, such that g is ¯ G -invariant.Let P ¯ G ( X, g ) be the set of C smooth ¯ G -invariant functions, such that ∀ φ ∈ P ¯ G ( X, g ) , ω + √− π ∂ ¯ ∂φ > and sup X φ = 0 . Then the ¯ G -invariant α -invariant of X is α ¯ G ( X ) = sup (cid:26) λ ∈ Q (cid:12)(cid:12)(cid:12)(cid:12) ∃ C ∈ R , such that ∀ φ ∈ P ¯ G ( X, g ) , R X e − λφ ω n ≤ C (cid:27) where n is the dimension of X . The number α ¯ G ( X ) was introduced in [24] and [25]. EAKLY-EXCEPTIONAL QUOTIENT SINGULARITIES 9
Definition 2.2.
Let X be a variety with at most Kawamata log termi-nal singularities (see [15, Definition 3.5] ) and D an effective Q -divisoron X . Let Z ⊆ X be a closed non-empty subvariety. Then the logcanonical threshold of D along Z is c Z ( X, D ) = sup { λ ∈ Q | the pair ( X, λD ) is log canonical along Z } To simplify notation, write c X ( X, D ) = c ( X, D ) . There exists an equivalent complex analytic definition of the logcanonical threshold:
Proposition 2.3 (see [15, Proposition 8.2]) . Let X be a smooth com-plex variety, Z a closed non-empty subscheme of X , and f a non-zeroregular function on X . Then c Z ( X, { f = 0 } ) = sup n λ ∈ Q (cid:12)(cid:12)(cid:12) | f | − λ is locally L near Z o Definition 2.4.
Let G be a finite subgroup of GL N ( C ) , where N ≥ ,and let ¯ G be its image under the natural projection into PGL N ( C ) .Then the global ¯ G -invariant log canonical threshold of P N − is: lct (cid:0) P N − , ¯ G (cid:1) = inf c (cid:0) P N − , D (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D is a ¯ G -invariant effective Q -divisor on P N − , such that D ∼ Q − K P N − Remark 2.5 (see [8, Theorem A.3]) . lct (cid:0) P N − , ¯ G (cid:1) = α ¯ G (cid:0) P N − (cid:1) Theorem 2.6 (see [6, Theorem 3.15]) . The singularity C N /G is weakly-exceptional ⇐⇒ lct( P N − , ¯ G ) ≥ . A similar condition is often necessary in order to compute conjugacyclasses in higher-dimensional Cremona groups. For details, see [4].
Theorem 2.7 (see [6, Theorem 3.18]) . Suppose that G ⊂ SL ( C ) isa finite group and ¯ G is its image under the natural projection intoPGL ( C ) . Then the following are equivalent: • the inequality lct( P , ¯ G ) ≥ holds, • the group G does not have semi-invariants of degree at most . Theorem 2.8 (see [6, Theorem 4.1]) . Suppose that G ⊂ SL ( C ) isa finite group and ¯ G is its image under the natural projection intoPGL ( C ) . Then the inequality lct( P , ¯ G ) ≥ holds if and only ifthe following conditions are satisfied: • the group G is irreducible, • the group G does not have semi-invariants of degree at most , • there is no ¯ G -invariant smooth rational cubic curve in P . Equivalently, the same criterion can be stated as:
Theorem 2.9.
Suppose that G ⊂ SL ( C ) is a finite group and ¯ G is itsimage under the natural projection into PGL ( C ) . Then the inequality lct( P , ¯ G ) ≥ holds if and only if the following conditions are satisfied: • the group G is irreducible, • the group G does not have semi-invariants of degree at most , • G is not a central extensions of A acting on C as the thirdsymmetric power of its irreducible -dimensional representa-tion.Proof: Assume ¯ G ⊂ PGL ( C ) is a finite irreducible subgroup thatdoes not fix any quadric or cubic surface, and C ⊂ P is a smoothrational ¯ G -invariant cubic curve.One can assume C is the image of C : ( x : y ) ∈ P (cid:0) x : x y : xy : y (cid:1) ∈ P This easily implies (by Proposition 2.11) that ¯ G must be isomorphicto one of the finite automorphism groups of P , with its action inducedby the action of P via C . This means ¯ G must be one of the following: • Cyclic or dihedral group: neither these groups nor their cen-tral extensions have an irreducible 4-dimensional representa-tion. Therefore such a G would not be irreducible. • A or S : The induced actions of these two groups preserve { ( x : y : u : v ) ∈ P | xy − uv = 0 } , which is a smooth quadricsurface. • A : This action is primitive. From the embedding of C into P ,it is easy to see that the action on C must correspond to ei-ther the third symmetric power of the irreducible 2-dimensionalrepresentation of A or its central extension. (cid:3) Remark 2.10.
The last condition of Theorem 2.9 is necessary, sincethis action is irreducible and does not preserve any projective surfacesof degree or (can be checked directly or seen in, for example, [6,Proof of Lemma 4.9] ). Properties of possible invariants.
Let ¯ G ⊂ PGL N ( C ) be afinite subgroup, and G be its lift to SL N ( C ). Assume that S ⊂ P N − is a proper ¯ G -invariant subvariety. Then there exists a natural homo-morphism π S : G → Aut ( S ) giving a short exact sequence0 −→ G −→ G π S −→ G S −→ G S = π S ( G ), G = ker π S . EAKLY-EXCEPTIONAL QUOTIENT SINGULARITIES 11
Proposition 2.11.
Let G ⊂ SL N ( C ) be a finite subgroup, and ¯ G the image of its natural embedding into PGL N ( C ) . Let S ⊂ P N − be asubvariety that is not contained in the union of any two proper linearsubspaces of P N − , and ¯ G fixes S point-wise. Then ¯ G is trivial.Proof: Pick g ∈ G . Then
Let G ⊂ SL N ( C ) and let ¯ G be its natural projectioninto PGL N ( C ) . Let S ⊂ P N − be a ¯ G -invariant subvariety, that is notcontained in the union of any two proper linear subspaces of P N − . Let π s : ¯ G → Aut ( S ) be the natural homomorphism. Then ker π S = 0 . Remark 2.13. If S ⊂ P N − is an irreducible surface, then either it iscontained in an ( N − -dimensional linear subspace of P N − or it isnot contained in the union of any two proper linear subspaces of P N − . Now assume further that G is irreducible. This implies the followingrestrictions: Remark 2.14.
In the notation above, the following hold: • ¯ G cannot be cyclic, as in that case G is abelian (as it is acentral extension by scalar elements), and all irreducible repre-sentations of abelian groups are -dimensional. • S cannot be contained in a proper linear subspace of P N − . Lemma 2.15.
In the notation above, let S have an isolated singularityof some given type (e.g. A n , D n , etc.). Then S has at least N isolatedsingularities of this type. Proof:
Let S have exactly k (1 ≤ k < N ) singularities of this type.Then they form a ¯ G -invariant set of k points, giving a k -dimensional¯ G -invariant subspace of P N . This is impossible, as G is irreducible. (cid:3) Three-dimensional case
Assume N = 3 throughout this section. This section is devoted toproving Theorem 1.14.The list of all finite subgroups of SL ( C ) is given by: Proposition 3.1 (see [26, Theorem A]) . Define the following matrices: S = ω
00 0 ω T = W = ω ω
00 0 ω U = ǫ ǫ
00 0 ǫω Q = a b c V = √− ω ω ω ω where ω = e πi/ , ǫ = ω and a, b, c ∈ C are chosen arbitrarily, as longas abc = − and Q generates a finite group.Up to conjugacy, any finite subgroup of SL ( C ) belongs to one of thefollowing types:(A) Diagonal abelian group.(B) Group isomorphic to an irreducible finite subgroups of GL ( C ) andnot conjugate to a group of type (A).(C) Group generated by the group in (A) and T and not conjugate toa group of type (A) or (B).(D) Group generated by the group in (C) and Q and not conjugate toa group of types (A)—(C).(E) Group of size generated by S , T and V .(F) Group of size generated by the group in (E) and an element P := U V U − .(G) Hessian group of size generated by the group in (E) and U .(H) Simple group of size isomorphic to alternating group A .(I) Simple group of size isomorphic to permutation group gener-ated by (1234567) , (142) (356) , (12) (35) .(J) Group of size generated by the group in (H) and W .(K) Group of size generated by the group in (I) and W .(L) Group G of size with its quotient G/
EAKLY-EXCEPTIONAL QUOTIENT SINGULARITIES 13
Remark 3.2.
The list of groups in Proposition 3.1 can be subdividedas follows: • Groups of types (A) and (B) are not irreducible, and so by The-orem 1.11 cannot induce a weakly-exceptional singularity. • Groups of types (E)—(L) are primitive. • Groups of types (C) and (D) are irreducible monomial.Proof:
Immediate from the lists of generators given above. (cid:3)
Let G ⊂ SL ( C ) be an irreducible finite subgroup with image ¯ G under the natural projection into PGL ( C ). Lemma 3.3.
Assume the singularity the action of G induces is notweakly-exceptional. Then: • ¯ G leaves a smooth curve C ⊂ P of degree invariant. • G is isomorphic to one of D n , A , S , A (some n ≥ ) or toone of their central extensions by scalar elements.Proof: By Theorem 2.7, ¯ G must preserve a curve C ⊂ P of degree2. If C is singular, then it must have exactly one isolated singularity,which is impossible by Lemma 2.15. Therefore, C must be smooth andhence rational, with ¯ G isomorphic to a finite irreducible subgroup ofAut ( P ). Proof of Theorem 1.14:
Groups of types (A) and (B) can be excludedimmediately, since they are not irreducible. Assume the singularity G induces is not weakly-exceptional. Then, comparing the lists inProposition 3.1 and Lemma 3.3, G must be conjugate to one of: • A central extension of ( Z ) ⋊ Z ∼ = A , where both Z -s act di-agonally and Z permutes the basis (such a group is of type (C)). • A central extension of ( Z ) ⋊S ∼ = S , where both Z -s act diag-onally and S permutes the basis (such a group is of type (D)). • A central extension of A (such a group is of type (H) or (J)).First consider the central extensions of A and S . Use their presen-tations from Proposition 3.1, and let ( x, y, z ) be the corresponding co-ordinates for C . Then the groups have a semi-invariant smooth conicdefined by x + y + z = 0. Similarly, it is easy to check (see [26, Sec-tion 2.9]) that the groups of type (H) have semi-invariant smooth con-ics. Since any group of type (J) is generated by a group of type (H) andthe scalar matrices, these groups will also have semi-invariant smoothconics.The statement of Theorem 1.14 follows by renaming the remaininggroups to fit with the more widely-used notation. (cid:3) Four-dimensional case
The aim of this section is to prove Theorem 1.15. By Theorem 1.11, if G is not irreducible, it cannot induce a weakly-exceptional singularity.Thus, one can restrict attention to the irreducible groups. Throughoutthis section, let G ⊂ SL ( C ) be a finite irreducible subgroup, and ¯ G its image under the natural projection SL ( C ) → PGL ( C ).In view of Theorem 2.9, the only irreducible ¯ G that are not weakly-exceptional are the finite subgroups of automorphism groups of surfacesof degree 2 or 3 and A with one specific action. When consideringautomorphisms of a surface S , without loss of generality one can assumethat there is no ¯ G -invariant surface S ′ of smaller degree. Lemma 4.1.
Let G ⊂ SL ( C ) be a finite irreducible group, ¯ G ⊂ PGL ( C ) its projection, and let S ⊂ P be a ¯ G -invariant surface ofdegree minimal among the degrees of all ¯ G -invariant surfaces. Theneither deg S ≥ or S is smooth.Proof: Since G is irreducible, deg S ≥
2. If deg S = 2 and S issingular, then either S has exactly 1 isolated singularity, or S is aunion of two planes and thus has a singular line, which must thenbe ¯ G -invariant. Both cases are impossible (by Lemma 2.15 and theirreducibility of G ), so if deg S = 2 then S must be smooth.If deg S = 3, and S is not irreducible, then either it is the union of aplane and an irreducible quadric surface (each of which must thus be a¯ G -invariant surface of smaller degree, contradicting the hypothesis) or S is the union of 3 distinct planes, whose intersection gives a point fixedby all of ¯ G (stopping G from being irreducible). Hence S is irreducible.Assume S has non-isolated singularities, with C being the union ofall singular curves on S . Then, one can easily see that C is a line.Since ¯ G ( S ) = S , ¯ G ( C ) = C , and so there exists a ¯ G -invariant line,contradicting irreducibility of G . Therefore if deg S = 3 then S musthave at worst isolated singularities.If deg S = 3 and S is singular with only isolated singularities, thenby [3], the singularity types form one of the following collections: ( A ),(2 A ), ( A , A ), (3 A ), ( A , A ), (2 A , A ), (4 A ), ( A , A ), (2 A , A ),( A , A ), ( A , A ). Since by Lemma 2.15, S cannot have exactly 1, 2or 3 singularities of any given type, S has to have 4 A singularities.Since there is only one such surface (see, for example, [3]), S must bethe Cayley cubic, defined (in some basis) by S = (cid:8) ( x : y : u : v ) ∈ P | xyu + xyv + xuv + yuv = 0 (cid:9) EAKLY-EXCEPTIONAL QUOTIENT SINGULARITIES 15
But in this case S contains exactly 9 lines, 6 of which are goingthrough pairs of singular points and the other 3 defined by x + y = 0 = u + v, x + u = 0 = y + v and x + v = 0 = y + u These last three lines are coplanar and must be mapped to each other byall of ¯ G . Therefore, ¯ G preserves a plane, contradicting the irreducibilityassumption for G . Thus if S is a cubic surface, then it must be smooth. (cid:3) Summarising, if ¯ G is irreducible but the singularity it induces is notweakly-exceptional, any ¯ G -invariant surface S of degree at most 3 mustbe a smooth surface of degree 2 or 3. These cases will be consideredseparately in the next two sections. Proof of Theorem 1.15:
By Theorem 2.9 and the discussion above,the theorem is an immediate consequence of Lemmas 4.2 and 4.3. (cid:3) If S is a smooth ¯ G -invariant cubic surface. This section isdevoted to proving the following lemma:
Lemma 4.2. If G ⊂ SL ( C ) is a finite irreducible subgroup, and ¯ G its projection to PGL ( C ) . Also assume that there is no ¯ G -invariantquadric surface, and S ⊂ P is a smooth ¯ G -invariant cubic surface.Then G must be isomorphic to a central extension of one of: • (cid:0) ( Z ) ⋊ Z (cid:1) ⋊ Z . This produces a monomial action. • ( Z ) ⋊ S . This produces a monomial action.by scalar elements, acting as described below. Comparisons with bothof these isomorphism classes are indeed necessary. As stated before, ¯ G ⊂ Aut ( S ) is a finite subgroup, so by [13], ¯ G must belong to one of the following isomorphism classes:(1) { e } , Z , Z , Z .(2) ( Z ) , S , S × Z .(3) S .(4) ( Z ) ⋊ Z .(5) (cid:0) ( Z ) ⋊ Z (cid:1) ⋊ Z .(6) S .(7) ( Z ) ⋊ S .4.1.1. Case ¯ G cyclic. The groups in (1) are all cyclic, so their exten-sions by scalar elements do not act irreducibly.4.1.2.
Case ¯ G dihedral. This case covers the isomorphism classes ( Z ) , S and S × Z . The dihedral groups and their extensions by scalarelements do not have any irreducible 4-dimensional representations, sothese groups cannot act irreducibly. Case ¯ G ∼ = S . This group by itself has no 4-dimensional irre-ducible representations, while its central extension has (up to a choiceof a root of unity) only one such, which preserves a quadric surface (seethe twisted diagonal actions in Lemma 4.3).4.1.4.
Cases ¯ G ∼ = ( Z ) ⋊ Z or (cid:0) ( Z ) ⋊ Z (cid:1) ⋊ Z . For convenience,write ¯ G ′ = ( Z ) ⋊Z and ¯ G ′′ = (cid:0) ( Z ) ⋊ Z (cid:1) ⋊Z , with all the notationfollowing in the obvious manner (i.e. write G ′ for the lift of ¯ G ′ toSL ( C ), etc.). Using the notation from [13], these two cases correspondto groups ¯ G ′ = G /C ( G ) and ¯ G ′′ = G ⋊Z . This means there existnon-trivial elements ¯ α, ¯ β, ¯ γ, ¯ δ, ¯ ǫ ∈ ¯ G ′′ , such that ¯ α, ¯ β, ¯ γ, ¯ δ generate G ,with ¯ α generating its centre, ¯ α = ¯ β = ¯ γ = ¯ δ = ¯ ǫ = id. Let α, β, γ, δ, ǫ be lifts of ¯ α, ¯ β, ¯ γ, ¯ δ (respectively) to SL ( C ).Let h := α , h := β . ¯ α, ¯ β commute, so say βα = αβh . By thestructure of the lift, h i are scalar matrices of order 1, 2 or 4. Then h h = (cid:0) α βα (cid:1) = ( βh h ) = h h h and so h = id. Similarly, get α, β, γ all commuting. Hence the cor-responding matrices can all be taken to be diagonal (by choosing asuitable basis). It is then easy to see that δ and ǫ must act as elementsof a central extension of S permuting the basis.Since ¯ G ′′ has only one normal subgroup of index 2, and ¯ G ′′ has nocentre (otherwise ¯ G ′′ /C (cid:0) ¯ G ′′ (cid:1) would be on the list of groups acting ona cubic surface), ¯ δ ¯ ǫ = ¯ ǫ ¯ δ . Therefore, up to conjugation, δ interchangesthe first and the second basis vectors, and ǫ interchanges the first basisvector with the third one and the second basis vector with the fourthone.This means that G ′ is not irreducible, while G ′′ is irreducible and(up to conjugation) is generated by ζ ζ − , ζ ζ − , ζ
00 0 0 ζ − ,ζ , This group leaves (for example) the cubic polynomial x + y + z + w (in coordinates ( x, y, z, w ) for C ) semi-invariant, and by directcomputation, one sees that the group does not have a semi-invariantquadric surface. This action of G ′′ is monomial. EAKLY-EXCEPTIONAL QUOTIENT SINGULARITIES 17
Case ¯ G ∼ = S . According to [21, § S = (cid:26) ( x : x : x : x : x ) ∈ P (cid:12)(cid:12)(cid:12)(cid:12) x + x + x + x + x = 0 ,x + x + x + x + x = 0 (cid:27) which immediately implies that there exists a ¯ G -invariant quadric sur-face. For the group’s action on it, see Lemma 4.3.4.1.6. Case ¯ G ∼ = ( Z ) ⋊ S . As stated in [21, § C arbitrarily and multiplying them byarbitrary cube roots of unity. Hence (up to conjugation) G is a centralextension of such a group by scalar elements.This group clearly leaves the cubic polynomial x + y + z + w (in coordinates ( x, y, z, w )) semi-invariant, and by direct computation,one sees that the group does not have a semi-invariant quadric surface.The action of this group is monomial.4.2. If S is a smooth ¯ G -invariant quadric surface. Let the group¯ G ⊂ PGL ( C ) be a finite irreducible subgroup, G its lift to SL ( C ) and S ⊂ P a smooth ¯ G -invariant quadric surface. This section is devotedto compiling a list of the possible values that G (equivalently, ¯ G ) cantake in this situation. The final list is presented in Lemma 4.3.In this case there exists a basis for P , in which S is the image of theSegre embedding of P × P into P . The image of this embedding is S = (cid:8) ( x : y : u : v ) ∈ P | xv − yu = 0 (cid:9) This implies that the subgroup of PGL ( C ) preserving S is isomorphicto (PGL ( C )) ⋊ Z , with the action that can be described in thenotation given in Sections 2.1 and 2.2. This notation will be usedthroughout the rest of the paper.Now one needs to put the action of any finite subgroup of Aut ( S )into one of the four categories: • Not irreducible. • Irreducible monomial. • Irreducible non-monomial imprimitive. • Primitive (hence irreducible).This can be done directly by looking at the representations used tobuild the action. To make the explanations more simple, it will beassumed that H and H (see Section 2.2) both contain a non-scalardiagonal matrix. This will mean that any proper invariant subspacemust have a basis, which is a subset of the chosen basis for C . It iseasy to check that for all the actions used in this section, there exists a basis for C in which the action contains such a matrix. Now fix thisbasis for the remainder of this section.4.2.1. Irreducibility:
Assume first that ¯ G does not interchange the rul-ings. Then it can be seen that in order for G to be irreducible, both H and H need to be irreducible: if H is not irreducible, then (in thematrix presentation of P ) the “rows” of P produce invariant lines, i.e. { x = y = 0 } , { u = v = 0 } ⊂ (cid:8) ( x : y : u : v ) ∈ P (cid:9) are invariant subspaces. Similarly, if H is not irreducible, then thereexist H -invariant lines corresponding to the “columns” of P , i.e. { x = u = 0 } , { y = v = 0 } ⊂ (cid:8) ( x : y : u : v ) ∈ P (cid:9) are invariant subspaces. Even when both H i are irreducible, G can stillfail to be irreducible, but this can only occur if both H i are dihedralgroups, with their order 2 elements coupled (see the end of Section 2.2).This means that for the action of G to be irreducible, need (for any i = j ∈ { , } )) one of the following to hold: • H i , H j ∈ { A , S , A }• H i ∈ { A , S , A } , H j = D n • H = D m , H = D n and the action contains elements of theform (cid:10) a l | b (cid:11) or (cid:10) b | a k (cid:11) , not just (cid:10) a l b | a k b (cid:11) ( k, l ∈ Z ).If ¯ G does interchange the rulings, then assume first that τ ∈ ¯ G (where τ is the involution interchanging the rulings, see Section 2.2).Then τ must keep one of the diagonals of the matrix form of P invari-ant, i.e. { x = v = 0 } , { y = u = 0 } ⊂ (cid:8) ( x : y : u : v ) ∈ P (cid:9) are left invariant by τ . However, if τ ∈ ¯ G , then H ∼ = H , so the onlytwo cases where it can potentially make a previously not irreducibleaction into an irreducible one are when H ∼ = H ∼ = D n (the actionis not irreducible, as (cid:10) a l b | a k b (cid:11) leaves the same two proper subspacesinvariant as τ does) or when H ∼ = H ∼ = Z n (the action is not irre-ducible, as τ only permutes pairs of coordinates, so the group has apair of distinct invariant proper subspaces)If ¯ G interchanges the rulings, but τ ¯ G , then στ ∈ ¯ G for someautomorphism σ that preserves the ruling, with σ H , σ ∈ H . Con-sider ˆ G generated by H , τ and σ , with ˆ K i , ˆ H i , ˆ H defined similarly to K i , H i , H . Clearly, ¯ G ⊂ ˆ G , H ⊂ ˆ H , etc. • If H i ∼ = Z m , then either ˆ H i ∼ = Z m or ˆ H i ∼ = D m with σ = (cid:10) a l b | a k b (cid:11) ∈ ˆ H . (some k, l ∈ Z ). In either case this makes ˆ G EAKLY-EXCEPTIONAL QUOTIENT SINGULARITIES 19 not irreducible, and hence ¯ G not irreducible (by the discussionabove). • If H i ∼ = D m , then it is easy to see that ˆ H i ∼ = D n (for some n > m ). Similarly to the considerations above, this only influ-ences the irreducibility of ¯ G if σ can be taken to be of the form (cid:10) a l b | a k (cid:11) , making the group action irreducible.The discussion above has been summarised in Table 1. Table 1.
Irreducibility of GH A , S , A D m Z m H A , S , A Irreducible Irreducible Not irreducible D m Irreducible Depends on action Not irreducible Z n Not irreducible Not irreducible Not irreducible4.2.2.
Primitivity:
Assume the action of G is irreducible. Again, thenotation from Section 2.2 will be used throughout. Then the questionof the action being monomial, imprimitive non-monomial or primitivebecomes of interest.By direct computation, it is easy to see that in most cases, the placeof G in this classification depends on the matrices in H i that have 3or more non-zero entries, and how these matrices are combined in G ,i.e. on the isomorphism α : H / ˆ K → H / ˆ K (as defined above). Theonly exception occurs when H i ∼ = D m , G interchanges the ruling, but τ ¯ G — in this case, the automorphism σ needs to be considered.With this in mind, direct computation provides the following criteria(putting i = j ∈ { , } ): • If H , H dihedral and G irreducible, then G acts monomially. • If H i ∈ { A , S , A } and H j = D n , then the action of G isnon-monomial imprimitive. • If H ∼ = H ∼ = A , then the action of G is primitive. • If H , H ∈ { A , S } and the 3-cycles are not coupled, then theaction of G is primitive. • If H ∼ = H ∼ = A and the 3-cycles are coupled, then the actionof G is monomial. • If H i ∼ = S , H j ∼ = A and the 3-cycles are coupled, then theaction is imprimitive non-monomial. • If H i ∼ = S , H j ∼ = S , the 3-cycles are coupled and the oddpermutations are coupled, then the action is monomial. • If H i ∼ = S , H j ∼ = S , the 3-cycles are coupled, but the oddpermutations are not coupled, then the action is primitive.This list is clearly not exhaustive, but it is sufficient for determiningthe nature of all the groups below.4.2.3. Possible isomorphism classes of ¯ G . Since ¯ G is a finite groupleaving a smooth quadric S invariant, its action must be equal (asshown in Section 2.4) to a suitable of one of the finite automorphismgroups of a smooth 2-dimensional quadric. Thus ¯ G must be conjugateto the image of one of the finite groups given in [9, Theorem 4.9].In order to make the structure of each of the groups slightly moreexplicit, the group structure will also be given in the notation( H , K , H , K ) α , where H i , K i are as before, and α is the gluing isomorphism between H / ˆ K and H / ˆ K . Where only one such isomorphism exists, α willbe omitted. For each isomorphism class, several representations of thegroup can be chosen. However, it is clear that the different faithfulrepresentations will differ by at most an outer automorphism, so allthe properties that are of interest in this discussion will be the samefor all of them. Therefore, for each isomorphism class, any faithfulrepresentation of H i can be chosen. For any group ( H , K , H , K ) α ,there also exists a group ( H , K , H , K ) α − , which corresponds to thesame group with the components of the ruling of the quadric swapped.These two groups are conjugate to each other. Lemma 4.3. If ¯ G ⊂ PGL ( C ) is a finite irreducible subgroup, G itslift to SL ( C ) and S ⊂ P a smooth ¯ G -invariant quadric surface, then ¯ G must be conjugate to one of the groups in the list below. From theway the groups are constructed, it is easy to see that all the groups doneed to be in the list. First assume ¯ G leaves the ruling of P × P invariant. Bearing theanalysis above in mind, for G to be irreducible, it must be conjugateto one of the following:(1) Product subgroups ( H , H , H , H ) ∼ = H × H for some finitegroups H i ∈ Aut ( P ). Taking different choices for H , H , getthe following groups of the form H × H :(a) 9 primitive groups when H , H ∈ { A , S , A } .(b) 3 families of non-monomial imprimitive groups D m × H ,where H ∈ { A , S , A } .(c) 1 family of monomial groups D m × D n . EAKLY-EXCEPTIONAL QUOTIENT SINGULARITIES 21 (2) Twisted diagonal subgroups ( H , , H , α ∼ = H for some finitegroup H ∈ Aut ( P ). This gives 3 families of groups, indexedby the choice of isomorphism α . They are:(a) Monomial groups isomorphic to A or S .(b) Primitive groups isomorphic to A .The twisted diagonal groups isomorphic to the dihedral groupsdo not act irreducibly, as the relevant central extensions do nothave any 4-dimensional irreducible representations.(3) [ S × S ] ∼ = ( A × A ) ⋊ Z ∼ = ( S , A , S , A ), a primitivegroup generated by elements corresponding to h [12][34] | id i , h id | [12][34] i , h [123] | id i , h id | [123] i and h (12) | (12) i .(4) [ D m × S ] ∼ = ( Z m × A ) ⋊ Z ∼ = ( D m , Z m , S , A ), a fam-ily of imprimitive non-monomial groups generated by h a m | id i , h id | [12][34] i , h id | [123] i and h b | (12) i .(5) [ D m × S ] ∼ = ( D m × A ) ⋊ Z ∼ = ( D m , D m , S , A ) ( m ≥ h a m | id i , h b | id i , h id | [12][34] i , h id | [123] i and h a m m | (12) i .(6) [ D m × S ] ∼ = ( Z m × V ) ⋊ S ∼ = ( D m , Z m , S , V ), a familyof imprimitive non-monomial groups generated by h a m | id i , h id | [12][34] i , h id | (13)(24) i , h a m m | [123] i and h b | (12) i .(7) [ S × S ] ∼ = ( V × V ) ⋊ S ∼ = ( S , V , S , V ), a monomialgroup generated by h [12][34] | id i , h (13)(24) | id i , h id | [12][34] i , h id | (13)(24) i , h [123] | [123] i and h (12) | (12) i .(8) [ A × A ] ∼ = ( V × V ) ⋊ Z ∼ = ( A , V , A , V ), a monomialgroup generated by h [12][34] | id i , h (13)(24) | id i , h id | [12][34] i , h id | (13)(24) i and h [123] | [123] i .(9) [ D m × D n ] ∼ = ( Z m × D n ) ⋊ Z ∼ = ( D m , Z m , D n , D n ) (for m, n ≥ h a m | id i , h id | a n i , h id | b i and h b | a n n i .(10) [ D m × D n ] α ∼ = ( Z m × Z n ) ⋊D ∼ = ( D m , Z m , D n , Z n ) α (where α ( b ) = a n n , α ( a m m ) = b ), a family of monomial groups generatedby h a m | id i , h id | a n i , h a m m | b i and h b | a n n i .(11) [ D m × D n ] ∼ = ( D m × D n ) ⋊ Z ∼ = ( D m , D m , D n , D n ) (for m, n ≥ h a m | id i , h b | id i , h id | a n i , h id | b i and h a m m | a n n i .Now assume that ¯ G does interchange the rulings (via an element σ ◦ τ ). Then the normal subgroup of ¯ G fixing the ruling of the quadricmust also be a group of automorphisms of S . Furthermore, it musthave H ∼ = H and K ∼ = K , since conjugation by στ provides the twoisomorphisms. That means that ¯ G can be isomorphic to one of thefollowing groups: (12) ( H × H ) ⋊ ∼ = ( H , H , H , H ) ⋊ Z ( H ∈ Aut ( P )). Takingdifferent choices for H and bearing in mind that choosing H to be Z n produces a group that is not irreducible (see discussionabove), get 2 families of monomial groups(a) ( D n × D n ) ⋊ Z and 3 families of primitive groups, all of them indexed by thepossible involutions acting on H :(b) ( A × A ) ⋊ Z .(c) ( S × S ) ⋊ Z .(d) ( A × A ) ⋊ Z .(13) H ⋊ Z ∼ = ( H , , H , α ( H ∈ Aut ( P )). This gives 3 familiesof groups, indexed by the choice of isomorphism α . They are:(a) Monomial groups isomorphic to D n ⋊ Z .(b) Monomial groups isomorphic to A ⋊ Z or S ⋊ Z .(c) Primitive groups isomorphic to A ⋊ Z .(14) (( A × A ) ⋊ Z ) ⋊ Z ∼ = ( S , A , S , A ) ⋊ Z , a family of prim-itive groups.(15) (( V × V ) ⋊ S ) ⋊ Z ∼ = ( S , V , S , V ) ⋊ Z , a family of mono-mial groups.(16) (( V × V ) ⋊ Z ) ⋊Z ∼ = ( A , V , A , V ) ⋊Z , a family of mono-mial groups.(17) (( Z m × Z m ) ⋊ D ) ⋊ Z ∼ = ( D m , Z m , D m , Z m ) α ⋊ Z (where α ( b ) = a m m , α ( a m m ) = b ), a family of monomial groups.(18) (( D m × D m ) ⋊ Z ) ⋊Z ∼ = ( D m , D m , D m , D m ) ⋊Z ( m ≥ Example 4.4.
When one thinks about finite groups in SL ( C ) thatleave a smooth quadric invariant, one of the first examples that cometo mind is the monomial group G = ( Z ) ⋊ S , where the normalsubgroup H = ( Z ) acts diagonally, and the symmetric group permutesthe basis. Here, one of the size two subgroups of H acts by scalarmatrices, so the action on the projective quadric surface is isomorphicto ( Z ) ⋊ S ∼ = (( V × V ) ⋊ Z ) ⋊ Z , which is in position (16) in thelist above. EAKLY-EXCEPTIONAL QUOTIENT SINGULARITIES 23
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