On quotients of M ¯ ¯ ¯ ¯ ¯ ¯ g,n by certain subgroups of S n
aa r X i v : . [ m a t h . AG ] A p r ON QUOTIENTS OF M g,n BY CERTAIN SUBGROUPS OF S n IRENE SCHWARZ
Abstract.
We show that certain quotients of the compactified moduli space of n − pointed genus g curves, M G := M g,n /G , are of general type, for a fairly broad class ofsubgroups G of the symmetric group S n which act by permuting the n marked points.The values of ( g, n ) which we specify in our theorems are near optimal in the sense that,at least in he cases that G is the full symmetric group S n or a product S n × . . . × S n m ,there is a relatively narrow transitional zone in which M G changes its behaviour frombeing of general type to its opposite, e.g. being uniruled or even unirational. As anapplication we consider the universal difference variety M g, n /S n × S n . introduction In this paper we shall consider a class of quotients of M g,n , the compactified modulispace of n -pointed genus g complex curves, by certain subgroups of the symmetric group S n which act by permuting the marked points. Our aim is to analyse under which con-ditions such quotients are of general type or, in a complementary case, uniruled or evenunirational. As usual, we do this by using the Kodaira dimension.We were led to considering the questions adressed in the present paper by analyzing ananalogous problem for the compactified moduli space N g,n of n − nodal genus g curves in[S]. Here N g,n = M g, n /G where the special group G is also a subgroup of S n , namelythe semidirect product G := ( Z ) n ⋊ S n . In view of the great importance of n − nodalcurves, e.g. in the deformation type arguments used in the proof of the Brill-Noethertheorem, this problem was directly motivated by geometry.Our proof in [S], however, let us realize that there are some related results for general quotients of M g,n which in some aspect are different from the special case of N g,n . Inparticular, it is important that G := ( Z ) n ⋊ S n is a semidirect product and not a productof subgroups. The main point of this paper is to prove first results in this direction for aclass of general quotients.For G ⊂ S n , we denote the quotient by this action as M Gg,n := M g,n /G and we suppressthe subscript ( g, n ) if we feel it unnecessary within a given context. Then the naturalquotients induce the chain of morphisms of schemes M g,n → M Gg,n → M S n g,n (1.1)which by standard arguments (weak additivity of the Kodaira dimension for base andfibre) gives the following ordering for the Kodaira dimension that κ ( M g,n ) > κ ( M Gg,n ) > κ ( M S n g,n ) . (1.2)Since all algebraic varieties in (1.1) have the same dimension, one gets
1N QUOTIENTS OF M g,n BY CERTAIN SUBGROUPS OF S n M g,n of general type ⇒ M Gg,n of g. t. ⇒ M S n g,n of g.t. (1.3)By the same argument, one gets Lemma 1.1.
For any subgroup H of G one has M G of general type ⇒ M H of general type (1.4)We shall need the ramification divisor R (see equation (2.9) below) of the quotient map π : M g,n → M G . Denoting by ( i j ) the transposition in S n interchanging i and j andchecking where sheets will come together, one readily finds R = X ( i j ) ∈ G δ , { i,j } (1.5)where the (standard) definition of the boundary divisor δ , { i,j } is given in Section 2 below,which in particular introduces all divisors needed in this paper. Then the well knownexplicit formula for the canonical divisor K M g,n gives Corollary 1.2.
The pullback K G := π ∗ ( K M G ) to M g,n is given by K G = K M g,n − R = 13 λ + ψ − δ − X ( i j ) ∈ G δ , { i,j } (1.6)For the sake of the reader, we shall now recall Proposition 1.3.
The moduli space M g,n is of general type for g > or for n > n min ( g ) given in the following table: g n min
16 15 16 15 14 13 11 12 11 11 10 10 9 9 9 7 6 4 4
Table 1.
This collects results from [L, F, FV4] and covers all cases known up to now. As acorollary, one then obtains
Theorem 1.4.
In all cases of Proposition 1.3, if G does not contain any transposition,then M G also is of general type.Proof. By Corollary 1.2, the divisor classes K G and K M g,n coincide. The moduli space M g,n is of general type if and only if K M g,n is the sum of an ample and an effective divisorand, similarly, M Gg,n is of general type if and only if K G is the sum of an ample and aneffective divisor which are both also G − invariant. But in all cases of Table 1 only S n − invariant divisors have been used to show that M g,n is of general type, see [L, F, FV4].This proves our claim. (cid:3) In particular, this covers the case where G is cyclic (and different from Z ) or thecardinality | G | is odd. It also covers the largest non-trivial subgroup of S n , the alternatinggroup A n = kernel signum (in fact G contains no transpositions, if and only if it is asubgroup of A n ). This has an obvious geometric interpretation: The set of n − tuples of n fixed different points on a genus g curve carries a notion of orientation : Two n − tupleshave the same orientation, if they are mapped one to another by an even permutation. N QUOTIENTS OF M g,n BY CERTAIN SUBGROUPS OF S n Taking the quotient by S n corresponds to passing from n − pointed curves to n − markedcurves, while taking the quotient by A n means passing to curves marked in n points withorientation. Under the first action the property of being of general type might change,but it is invariant under the second.We remark that it is very natural to use S n − invariant divisors. We expect that it ispossible to do so in any case in which M g,n is of general type. Thus we conjecture: If M g,n is of general type, then M G is of general type for any subgroup G of A n .If the subgroup G does contain transpositions, the situation is more complicated. Forlarge g , the following theorem contains an (easy) general result, while for small g weshall need explicit computations with well chosen divisors. Collecting from [FV2] andsupplementing this by an analog proof in the cases g = 22 ,
23 (using the divisors of[FV2]) we obtain
Proposition 1.5.
The space M S n (and thus, by Lemma 1.1, the space M G for anysubgroup G of S n ) is of general type if (i) g > , n < g, or (ii) 13 g , and n min ( g ) n g − , where n min ( g ) is given in the followingtable g
12 13 14 15 16 17 18 19 20 21 22 23 n min
10 11 10 10 9 9 10 7 6 4 7 1
Table 2.
Furthermore, the domain of values ( g, n ) for which M S n is of general type is nearoptimal since it is known that (1) uniruled, if n > g (for any g ) or g ∈ { , } with n = g (2) unirational, if g < , n g (3) for g > the Kodaira dimension κ ( M g,g ) = 3 g − is intermediary. Here the result of (i) follows from weak additivity of the Kodaira dimension, while (ii) isproven in [FV2] by explicit computation. The first assertion in (1) follows from Riemann-Roch, while the second is proved in [FV1] which also contains (2) and (3). We briefly recallthe Riemann-Roch argument: Observe that the fibre of M g,n /S n → M g over a smoothcurve [ C ] ∈ M g is birational to the symmetric product C n := Sym n ( C ) := C n /S n . Thiscan be interpreted as the space of effective divisors of degree n on the curve C . Since theRiemann-Roch theorem implies that any effective divisor of degree d > g lies in some g d ,the quotient M g,n /S n is trivially uniruled for n > g .Thus, outside the domain of values specified in Proposition 1.5, there is just a narrowtransitional band in which M S n changes from general type to its opposite.It is, however, left open by Proposition 1.5 if M G might be of general type for somevalues of n > g if the subgroup G of S n is chosen judiciously. In the following theoremwe shall display a large class of groups for which this is the case. Theorem 1.6.
Fix a partition n = n + . . . + n m and let G = S n × . . . × S n m . Then M Gg,n (and thus, by Lemma 1.1, M Hg,n for any subgroup H of G ) is of general type if (i) g > , max { n , . . . n m } g − or (ii) g , max { n , . . . n m } g − and f m ( g ; n , . . . n m ) , where f m is thefunction defined in equation (4.39) of Section 4 below. N QUOTIENTS OF M g,n BY CERTAIN SUBGROUPS OF S n (iii) g , max { n , . . . n m } g − and f m ( g ; n , . . . n m , L , . . . , L m ) , where f m is the function defined in equation (4.44) of Section 4 below (depending on achoice of divisor classes L , . . . , L m as described at the end of Section 4.Furthermore, M G still has non-negative Kodaira dimension if max { n , . . . n m } g and f m ( g ; n , . . . n m ) . A geometric interpretation of this result is similar to the interpretation for A n : Thepartition P : n = n + . . . + n m induces the group G = G P = S n × . . . × S n m , and the actionof G P maps an n -pointed genus g curve to a curve with markings in n , . . . , n m (consideredas an ordered m − tuple) which we may call a P − marked curve. Thus Theorem 1.6 statesthat the moduli space of P − marked genus g curves is of general type (if the conditionsin Theorem 1.6 are satisfied).Since there is no upper bound on the number of summands m in the partition of n , an inspection of the defining equation for f m shows that the values of n may tendto infinity, provided the subgroup G is chosen appropriately. As in the above case for m = 2 , Riemann-Roch establishes a (small) transitional band beyond which M Gg,n becomesuniruled. We emphasize that the existence of this transitional band (for any fixed subgroup G ) is different from the result for N g,n proved in [S]: N g,n = M g, n / ( Z ) n ⋊ S n is of generaltype for all values of n , if g >
24. This is perfectly compatible with Theorem 1.6, since G := ( Z ) n ⋊ S n with its action on the 2 n marked points is not given by a direct productsubgroup of S n , as required in Theorem 1.6. To understand this from a more conceptualpoint of view, however, is wide open at present. We emphasize that it is not merelythe size of the group which is relevant: The alternating group G = A n might be takenarbitrarily large and still M G will preserve general type, while taking the quotient bymuch smaller groups, e.g. G = S g +1 will turn M G to being uniruled.As an application we consider m = 2 and the special case G = S n × S n . This quotienthas a geometric interpretation as the universal difference variety, i.e. the fibre of themap M G → M g over a smooth curve C is birational to the image of the difference map C n × C n → J ( C ) , ( D, E ) D − E , see e.g. [ACGH].Then M Gg, n is uniruled for n > g (by the Riemann-Roch argument from above, com-bined with the fact that a product is uniruled if one factor is), while Theorem 1.6 givesthe following. Proposition 1.7.
The universal difference variety M g, n /S n × S n is of general type for g > , n g − , or, in the low-genus case, if g and n min ( g ) n g − where n min ( g ) is specified in the following table g
10 11 12 13 14 15 16 17 18 19 20 21 22 23 n min Table 3.
This corollary amplifies the results of [FV3], which considers the universal differencevariety in the special case n = ⌈ g ⌉ . We emphasize that the result in Table 3 for g = 13and n = 7 is taken from [FV3]; in view of the sharp coupling between g and n they areable to use in this special case an additional divisor, which is not applicable in the othercases and which is not contained in our Section 3. All other cases in Table 3 follow fromour Theorem 1.6. N QUOTIENTS OF M g,n BY CERTAIN SUBGROUPS OF S n The outline of the paper is as follows. In Section 2 we introduce notation and somepreliminary results, in Section 3 we introduce the class of divisors used in our proof. Herewe basically recall, for the sake of the reader, some material from [S]. In Section 4 weprove Theorem 1.6. The use of a small program in computer algebra is appropriate tocheck our calculations. 2.
Preliminaries and Notation
The aim of this section is to develop a sufficient condition for M S n being of generaltype. This requires a basic understanding of the Picard group Pic( M S n ) and an explicitdescription of the boundary divisors and tautological classes on M S n which we shallalways consider as S n -invariant divisors on M g,n (any such divisor descends to a divisoron M S n ). For results on M g,n we refer to the book [ACG] (containing in particular therelevant results from the papers [AC1] and [AC2]). All Picard groups are taken withrational coefficients and, in particular, we identify the the Picard group on the modulistack with that of the corresponding coarse moduli space.In particular, we recall the notion of the Hodge class λ on M g,n , which automaticallyis S n − invariant and thus gives the Hodge class λ on M S n (where, by the usual abuse ofnotation, we denote both classes by the same symbol; this abuse of notation is continuedthroughout the paper).In order to describe the relevant boundary divisors on M g,n , we recall that ∆ (some-times also called ∆ irr ) on M g is the boundary component consisting of all (classes of)stable curves of arithmetical genus g , having at least one nodal point with the prop-erty that ungluing the curve at this node preserves connectedness. Furthermore, ∆ i , for1 i ⌊ g ⌋ , denotes the boundary component of curves possessing a node of order i (i.e.ungluing at this point decomposes the curve in two connected components of arithmeticalgenus i and g − i respectively). Similarly, on M g,n and for any subset S ⊂ { , . . . , n } ,we denote by ∆ i,S , i ⌊ g ⌋ , the boundary component consisting of curves possessinga node of order i such that after ungluing the connected component of genus i containsprecisely the marked points labeled by S . Note that, if S contains at most 1 point, onehas ∆ ,S = ∅ (the existence of infinitely many automorphisms on the projective line tech-nically violates stability). Thus, in that case, we shall henceforth consider ∆ ,S as thezero divisor.We shall denote by δ i , δ i,S the rational divisor classes of ∆ i , ∆ i,S in Pic M g and Pic M g,n ,respectively. Note that δ is also called δ irr in the literature, but we shall reserve thenotation δ irr for the pull-back of δ under the forgetful map π : M g,n → M g .We write δ for the sum of all boundary divisors and set δ i,s = P | S | = s δ i,S .We remarkthat a single δ i,S is not G − invariant (for a subgroup G of S n ), but the divisor P g ∈ G δ i,g ( S ) ,averaged by the action of G , obviously is. In particular δ and δ i,s are always G -invariant.We shall use such an averaging in the proof of Theorem 1.6.Next we recall the notion of the point bundles ψ i , i n, on M g,n . Informally, theline bundle ψ i (sometimes called the cotangent class corresponding to the label i ) is givenby choosing as fibre of ψ i over a point [ C ; x , . . . , x n ] of M g,n the cotangent line T vx i ( C ).For later use we also set ω i := ψ i − X S ⊂{ ,...,n } ,S ∋ i δ ,S , (2.7)and introduce ψ = P ni =1 ψ i . Clearly, the class ψ is S n − invariant. N QUOTIENTS OF M g,n BY CERTAIN SUBGROUPS OF S n As a first step in the direction of our sufficient criterion we need the following result onthe geometry of the moduli space M G . Theorem 2.1.
For any subgroup G of S n , the moduli space M G has only canonical singu-larities or in other words: The singularities of M G do not impose adjunction conditions ,i.e. if ρ : ˜ M G → M G is a resolution of singularities, then for any ℓ ∈ N there is anisomorphism ρ ∗ : H (( M G ) reg , K ⊗ ℓ ( M G ) reg ) → H ( ˜ M G , K ⊗ ℓ ˜ M G ) . (2.8) Here ( M G ) reg denotes the set of regular points of M G , considered as a projective varietyand K ˜ M G , K ( M G ) reg denote the canonical classes on ˜ M G and ( M G ) reg . The proof follows the lines of the the proof of Theorem 1.1 in [FV1]. We shall brieflyreview the argument. A crucial input is Theorem 2 of the seminal paper [HM] whichproves that the moduli space M g has only canonical singularities. The proof relies on theReid-Tai criterion: Pluricanonical forms (i.e. sections of K ⊗ ℓ ) extend to the resolution ofsingularities, if for any automorphism σ of an object of the moduli space the so-called age satisfies age ( σ ) >
1. The proof in [FV1] then proceeds to verify the Reid-Tai criterion forthe quotient of M g,n by the full symmetric group S n . Here one specifically has to considerthose automorphisms of a given curve which act as a permutation of the marked points.For all those automorphisms the proof in [FV1] verifies the Reid-Tai criterion. Thus, inparticular, the criterion is verified for all automorphisms which act on the marked pointsas an element of some subgroup of S n . Thus, the proof in [FV1] actually establishes theexistence of only canonical singularities for any quotient M g,n /G where G is a subgroupof S n . Clearly, this is our theorem.Theorem 2.1 implies that the Kodaira dimension of M G equals the Kodaira-Iitakadimension of the canonical class K M G . In particular, M G is of general type if K M G isa positive linear combination of an ample and an effective rational class on M G . It isconvenient to slightly reformulate this result. We need Proposition 2.2.
The class ψ on M g,n is the pull-back of a divisor class on M G whichis big and nef.Proof. Farkas and Verra have proven in Proposition 1.2 of [FV2] that the S n -invariant class ψ descends to a big and nef divisor class N g,n on the quotient space M g,n /S n . Considerthe sequence of natural projections M g,n π −→ M G ν −→ M S n . Then ν ∗ ( N g,n ) is a big and nefdivisor on M G = M g,n /G and π ∗ ( ν ∗ ( N g,n ) = ψ . (cid:3) Now observe that the ramification divisor (class) of the quotient map π : M g,n → M G is precisely R = X ( i,j ) ∈ G δ , { i,j } . (2.9)In fact ramification requires existence of a non-trivial automorphism belonging to G , andby standard results this only occurs in the presence of the projective line with 2 markedpoints that can be swapped. The non-trivial automorphism is then the transposition (ofthe labels) of these two marked points. Furthermore, the Hurwitz formula for the quotient N QUOTIENTS OF M g,n BY CERTAIN SUBGROUPS OF S n map π gives K := π ∗ ( K M G ) = K M g,n − R = 13 λ + ψ − δ − X ( i,j ) ∈ G δ , { i,j } . (2.10)We thus obtain the final form of our sufficient condition: If K is a positive multiple of ψ + some effective G − invariant divisor class on M g,n , then M G is of general type.3. divisors In this section we introduce the relevant S n − invariant effective divisors on M g,n . Firstwe recall the following standard result.
Proposition 3.1.
Let f : X → Y be a morphism of projective schemes, D ⊂ Y be aneffective divisor and assume that f ( X ) is not contained in D . Then f ∗ ( D ) is an effectivedivisor on X . In our case the assumption of this proposition is fulfilled automatically since we onlyconsider surjective maps.We shall need invariant divisors on M g,n . Rather than exhibiting them directly byexplicit definitions, we shall simply recall from the literature the existence of specialdivisors with small slope : If g + 1 is not prime, then there is an effective S n − invariantdivisor class D on M g (of Brill-Noether type) of slope s ( D ) = 6 + 12 g + 1 , (3.11)while for g +1 odd (which trivially includes the case g +1 being prime) there is an effective S n − invariant divisor class D on M g (of Giesecker-Petri type) of slope s ( D ) = 6 + 14 g + 4 g + 2 g , (3.12)see [EH]. For a few cases ( g = 10 , , ,
21) it has been shown in [FV4] (for g=12) and[F] (for the other 3 cases) that there exist special effective invariant divisors D = D g witheven smaller slope, i.e. s ( D g ) = g = 106 + g = 126 + g = 166 + g = 21 . (3.13)We shall need them in the proof of Theorem 1.5.Finally, we need divisors of Weierstrass-type, and these we have to introduce explicitly.We recall from [L], Section 5, the divisors W ( g ; a , . . . , a m ) on M g,m , where a i > P a i = g . They are given by the locus of curves C with marked points p , . . . , p m suchthat there exists a g g on C containing P i m a i p i . We want to minimize the distancebetween the weights a i . Thus we decompose g = km + r , with r < m , and set˜ W g,m = W ( g ; a , . . . , a m ) , a j = k + 1 (1 j r ) , a j = k ( r + 1 j m ) . (3.14) N QUOTIENTS OF M g,n BY CERTAIN SUBGROUPS OF S n This gives, in view of [L], Theorem 5.4,˜ W g,m = − λ + r X i =1 ( k + 1)( k + 2)2 ω i + m X i = r +1 k ( k + 1)2 ω i − · δ irr − X i,j r ( k + 1) δ , { i,j } − X i r,j>r k ( k + 1) δ , { i,j } − X i,j>r k δ , { i,j } − higher order boundary terms , (3.15)where higher order means a positive linear combination of δ i,S where either i > | S | > W g,m we want to generate a G − invariant divisor class ˜ W g,n,m on M g,n , by sum-ming over appropriate pullbacks. Thus we let S, T be disjoint subsets of { , . . . , n } with | S | = r and | T | = m − r (recall that r is fixed by the decomposition g = mk + r ) and let π S,T : M g,n → M g,m (3.16)be a projection (i.e. a surjective morphism of projective varieties) mapping the class[ C ; q , . . . , q n ] to [ C ; p , . . . , p m ] , where the points q i labeled by S are sent to the points p , . . . , p r (all with weights a i = k + 1) and the points labeled by T are sent to the points p r +1 , . . . , p m (all with weights equal to k ). Clearly, for fixed g , there are precisely (cid:0) nr (cid:1)(cid:0) n − rm − r (cid:1) such projections. With this notation, we introduce˜ W g,n,m := X S,T π ∗ S,T ˜ W g,m = − w λ λ + w ψ ψ + 0 · δ irr − X s > w s δ ,s − higher order boundary terms , (3.17)where higher order denotes a positive linear combination of boundary divisors δ i,S with i > w s > sw ψ > w ψ for s > , (3.18) w λ = (cid:18) nr (cid:19)(cid:18) n − rm − r (cid:19) , (3.19) w ψ = (cid:18) n − r − (cid:19)(cid:18) n − rm − r (cid:19) ( k + 1)( k + 2)2 + (cid:18) n − r (cid:19)(cid:18) n − r − m − r − (cid:19) k ( k + 1)2 , (3.20) w =2 w ψ + (cid:18) n − r − (cid:19)(cid:18) n − rm − r (cid:19) ( k + 1) + 2 (cid:18) n − r − (cid:19)(cid:18) n − r − m − r − (cid:19) k ( k + 1)+ (cid:18) n − r (cid:19)(cid:18) n − r − m − r − (cid:19) k . (3.21)Equation (3.20) is proved by applying pullback to (3.15), using ω := P ni =1 ω i , X S,T r X i =1 π ∗ S,T ω i = (cid:18) nr (cid:19)(cid:18) n − rm − r (cid:19) rn ω = (cid:18) n − r − (cid:19)(cid:18) n − rm − r (cid:19) ω (3.22)and X S,T m X i = r +1 π ∗ S,T ω i = (cid:18) nr (cid:19)(cid:18) n − rm − r (cid:19) m − rn ω = (cid:18) n − r (cid:19)(cid:18) n − r − m − r − (cid:19) , (3.23) N QUOTIENTS OF M g,n BY CERTAIN SUBGROUPS OF S n noting that equation (2.7) implies ω = ψ − X S | S | δ ,S . (3.24)The sums over the pullbacks of the boundary divisors are computed by similar combi-natorial considerations which we leave to the reader. Note that both the summand 2 w ψ on the right hand side of (3.21) and the bound in (3.18) are generated by the change ofbasis given in (2.7).Next, for the proof of Theorem 1.6 it will be convenient to renormalize the divisor ˜ W g,n,m in such a way that the coefficient of ψ is equal to 1. We thus introduce W g,n = w ψ ˜ W g,m and find, setting m = min { g, n } , W g,n = a ( g, n ) λ + ψ + 0 · δ irr − X s > b s δ ,s − higher order boundary terms , (3.25)where higher order denotes a positive linear combination of boundary divisors δ i,S with i > a ( g, n ) = ( n ( k +1)( g + r ) g = kn + r, r < n ng n > g (3.26) b = b ( g, n ) with b ( g, n ) = ( n − r ( r − k +1) +2 r ( n − r ) k ( k − r )+( n − r )( n − r − k r ( k +1)( k +2)+( n − r ) k ( k +1) g = kn + r, r < n g − n − n > g (3.27)and b s > b for all s > T g on M g,g − satisfying T g = − g − g − λ + ψ − g − δ irr − (cid:18) g − (cid:19) δ , + h.t. (3.28)where the higher order terms h.t. denote a linear combination of all other boundarydivisors with coefficients − g > m g/ F g,m on M g,n (with n = g − m )satisfying F g,m = aλ + ψ − b irr δ irr − b , δ , + h.t., (3.29)where, as above, the higher order terms h.t. denote a linear combination of all otherboundary divisors with coefficients − a = nn − (cid:18) mg − − gg − m (cid:19) , b , = 3 + ( g − n )( n + 1)( g + n )( n − , b irr = nm ( g − n − g and n have different parity, we set n = g − m + 1and pull back F g,m given in eqution 3.29 in all possible ways to M g,n (via a forgetfulmap forgetting one of the marked points). Summing all these divisor classes and thennormalizing gives an effective divisor class ˜ F g,m on M g,n satisfying˜ F g,m = aλ + ψ − b irr δ irr − b , δ , + h.t., (3.31) N QUOTIENTS OF M g,n BY CERTAIN SUBGROUPS OF S n where, as above, the higher order terms h.t. denote a linear combination of all otherboundary divisors with coefficients − a = nn − (cid:18) mg − − gg − m (cid:19) , b , = 3 + g − n − g + n − , b irr = nm ( g − n − . (3.32)4. Proof of Theorem 1.6
For assertion (i), recall that M g is of general type for g > . Furthermore, the genericfibre of the canonical projection M G → M g is C n /G ≃ ( C n /S n ) × . . . × ( C n m /S n m ) . For max { n , . . . , n m } g − C n i /S n i is of general type, and thus is the product C n /G. Therefore the assertion follows from weak additivity of the Kodaira dimension.Assertion (ii) is more complicated, involving divisors. We take Weierstrass divisors W k = W g,n k on each M g,n k and W = W g,n on M g,n , with coefficients a ( g, n k ) , b ( g, n k ) and a ( g, n ) , b ( g, n ) respectively, see Section 3 (3.25). Let S k be the set of points correspondingto the summand n k in the partition n = P k m n k , and denote by π k : M g,n → M g,n k the forgetful map forgetting all points except those in S k . In order to calculate π ∗ k W k weintroduce some notation.For any sets S ⊂ { , . . . , n } we define δ S,ℓi,s := X | T ∩ S | = ℓ, | T | = s δ i,T (4.33)and denote by π S : M g,n → M g, | S | the natural forgetful map. With this notation, by theusual abuse of notation amplified in Section 2, we have Proposition 4.1.
The pull-back divisors are π ∗ S ( λ ) = λ, π ∗ S ( δ irr ) = δ irr and π ∗ S ( ψ ) = X i ∈ S ψ i − n − n k +1 X s =2 δ S, ,s , π ∗ S ( δ i,s ) = X ℓ > δ S,si,s + ℓ . (4.34)Furthermore, observe that the labels i, j belong to different components S k , S ℓ if andonly if the transposition ( i j ) is not in G . This gives: the divisor L := P k m π ∗ k W k hasthe decomposition L = − X k m a ( g, n k ) λ + ψ − X ( i j ) / ∈ G δ , { i,j } + 0 δ irr − X k m b ( g, n k ) X i,j ∈ S k δ , { i,j } + h.t, (4.35)where h.t. denotes a (higher order) sum of boundary divisors, each multiplied with coef-ficients < −
2. In addition we consider W = − a ( g, n ) λ + ψ − b ( g, n ) X i,j δ , { i,j } + h.t, D = sλ − δ irr + h.t., (4.36)where D = D g is chosen with minimal slope s = s ( g ) (see the list of divisors with smallslope in (3.11)-(3.13)) and set ǫ := min { b ( g, n k ) − | k ∈ { , . . . , m } with n k > } . (4.37)Clearly, ǫ > { n , . . . n m } g −
2. Combining equations (4.35),(4.36), (4.37) (see also 3.25) one obtains the decomposition K G > D + 11 + ǫ L + 2 ǫb ( g, n )(1 + ǫ ) W + ηψ, η := ǫ ǫ (1 − b ( g, n ) ) > , (4.38) N QUOTIENTS OF M g,n BY CERTAIN SUBGROUPS OF S n provided one has the inequality f m ( g, n , . . . , n m ) := 2 s ( g ) −
11 + ǫ X k m a ( g, n k ) − ǫb ( g, n )(1 + ǫ ) a ( g, n ) . (4.39)Since by Proposition 2.2 the divisor class ψ is big and nef and all divisors in equation(4.38) are effective and S n − invariant, the proof boils down to checking the inequality(4.39).Note that for max { n , . . . , n m } ∈ { g − , g } we get ǫ = η = 0, which proves that K G isat least effective and thus gives non-negative Kodaira dimension.To treat the additional case max { n , . . . n m } = g − L , . . . , L m . The function f m in equation (4.39) will then depend on these divisors,destroying the explicit form of f m given in equation (4.39).Instead of the family of (generalized) Weierstrass divisores W k on M g,n k , for 1 k m, we use divisors L k on M g,n k , for 1 k m, having a decomposition L k = a k λ + ψ − b k,irr δ irr − b k δ , + h.t. (4.40)where b k > h.t. denote a linear combination of all otherboundary divisors with coefficients −
2. Setting (analog to the above) L := P k m π ∗ k L k we obtain L = − X k m a k λ + ψ − X ( i j ) / ∈ G δ , { i,j } − X k m b k,irr δ irr − X k m b k X i,j ∈ S k δ , { i,j } + h.t, (4.41)with h.t. as above. This is analog to (4.35).In this notation, we already have for shortness’s sake suppressed dependence on g, n .Using the same convention in equation (4.36) - thus simply writing a, b in the decompo-sition of W - and introducing ǫ := min { b k − | k ∈ { , . . . , m } with n k > } , (4.42)which is (4.37) with b ( g, n k ) replaced by b k and writing α + := max { α, } , we obtain thedecomposition K G > (2 −
11 + ǫ X k b k,irr ) + D + 11 + ǫ L + 2 ǫb (1 + ǫ ) W + ηψ, where η := ǫ ǫ (1 − b ) > , (4.43)provided one has the inequality f m ( g, n , . . . , n m , L , . . . , L m ) := (2 −
11 + ǫ X k b k,irr ) + s + 11 + ǫ X k m a k − ǫb (1 − ǫ ) a . (4.44)This finishes the proof. 5. Proof of Proposition 1.7
In case n g − n min (20) = 5 (instead of 4) and n min (22) = 6 (instead of 5) thisis a direct application of Theorem 4, case (ii).To cover the remaining cases, we shall use Theorem 4 in case case (iii) with the followingchoice of divisors L k for 1 k m = 2 for the difference variety) in equation(4.44).In case n = g −
1, we choose L = L = T g , defined in equation (3.28). It is straightfor-ward to check that f m
13 in this case.
N QUOTIENTS OF M g,n BY CERTAIN SUBGROUPS OF S n In case g = 20 and n = 4, we choose L = L = F , . Again, since now all divisorclasses are explicit, it is straightforward to check f m
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