On the Kodaira dimension of the moduli space of nodal curves
aa r X i v : . [ m a t h . AG ] M a y ON THE KODAIRA DIMENSION OF THE MODULI SPACE OFNODAL CURVES
IRENE SCHWARZ
Abstract.
We show that the compactification of the moduli space of n − nodal curvesof geometric genus g , i.e. N g,n := M g, n /G , with G := ( Z ) n ⋊ S n , is of general type for g >
24, for all n ∈ N . While this is a fairly easy result, it requires completely differenttechniques to extend it to low genus 5 g
23. Here we need that the number of nodesvaries in a band n min ( g ) n n max ( g ), where n max ( g ) is the largest integer smallerthan (or in some cases equal to) ( g − −
3. The lower bound n min ( g ) is close to thebound found in [L], [F1] for M g, n to be of general type (in many cases it is identical).This will be tabled in Theorem 1.1 which is the main result of this paper. Introduction
The goal of this paper is to study the moduli space N g,n := M g, n /G of n − nodal curvesof geometric genus g and its compactification N g,n := M g, n /G . Here M g, n is the modulispace of smooth curves of genus g with 2 n distinct marked points on which the group G := ( Z ) n ⋊ S n , with S n being the symmetric group, acts in the following way: Wegroup the labels { , ..., n } in pairs ( i, n + i ) , where i n , corresponding to pairs ofpoints ( x i , y i ) (where y i = x i + n ). Each of the n copies of Z acts on one of these pairsby switching components, and S n acts by permutation of the pairs. Then G acts on themoduli space M g, n by acting on the labels of the 2 n marked points. We shall henceforthassume without further comment that this grouping is fixed whenever we consider 2 n marked points. Clearly, G then acts transitively on the marked points and fixed pointfree on the moduli space.A special interest in nodal curves, and thus in the study of the moduli space N g,n , (withthe aim of better understanding the general properties of smooth curves) is a commonfeature of all deformation type arguments. They go back at least to Severi who proposedin [S] to use g − nodal curves to prove the Brill-Noether theorem, see [BN]. Such defor-mation type methods have become prominent in modern algebraic geometry, e.g. in thesystematic theory of limit linear series (see [EH3]) which progressively simplified the ear-lier deformation type arguments in the first rigorous proof of the Brill-Noether theoremand the Gieseker-Petri theorem (see [GH], [G], [EH1], [EH2],[Sch]).Thus it seems natural to study the moduli space N g,n of nodal curves in its own right.A crucial point in our analysis is the following diagram (which, incidentally, also showsthat N g,n actually parametrizes n − nodal curves with geometric genus g and arithmeticgenus g + n ):
1N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 2 M g, n M g + n N g,n π χµ which factors χ through the quotient map π , χ being the map which glues x i to y i forevery pair ( x i , y i ). Thus each pair is transformed into a nodal point of the curve C underconsideration, increasing its arithmetic genus by n .While it was known for a long time that the moduli space M g of algebraic curves isuniruled for low genus g , Eisenbud, Harris and Mumford showed in the 1980’s that M g isof general type for g >
24 (see [HM] and [EH4]), the case of M being somewhat special(see also [F2]).Since then there has been a lot of research refining this picture of the birational geom-etry of moduli spaces of curves. The addition of marked points on the algebraic curveleads to the moduli space M g,n of n -pointed curves of genus g whose Kodaira dimensionwas studied in [L], with some improvements in [F1], employing refined versions of theoriginal techniques. Heuristically, additional marked points push the moduli space in thedirection of being of general type. In particular, M ,n is of general type for all n > g
22 the moduli space M g,n achieves general type if n > n min ( g )for n min ( g ) sufficiently large. In another direction, the paper [BFV] studies the Kodairadimension of the Picard variety Pic dg parametrizing line bundles of degree d on curves ofgenus g .A further gratifying aspect of N g,n = M g, n /G is its structure as a quotient of thewell-understood space M g, n by a finite group. Such quotients have been studied beforein various contexts, and their birational geometry might be different from what one couldnaively expect.For instance, the space C g,n := M g,n /S n , for g large,is of general type for n g − n > g + 1; only the transitional case g = n is challenging (see [FV1]). In fact,to see this for large n , one observes that the fibre of M g,n /S n → M g over a smooth curve[ C ] ∈ M g is birational to the symmetric product C n . Since the Riemann-Roch theoremimplies that any effective divisor of degree d > g lies in some g d , the quotient M g,n /S n istrivially uniruled for n > g .This proof, of course, uses the very special structure of the above fibre as a symmetricproduct C n (which can be interpreted as a family of divisors on C and thus is accessiblevia Brill-Noether theory) and not just abstract properties of the group S n . Still, it pointsto the possibility that taking the quotient with respect to a sufficiently large group mightsomehow destroy the property of an algebraic variety of being of general type. At least forlow genus, this is compatible with the findings of this paper, see Theorem 1.1 below. Inour case, such a phenomenon might be related to the existence of an upper finite bound n max ( g ) for the number of nodes allowed on the geometric genus g curve Note that ourgroup G ⊂ S n is a subgroup of S n , implying that M g, n → M g, n /S n factors through N g,n . Thus the space N g,n should be somewhat intermediary between M g, n and C g, n .Making this precise is the central result of the present paper. Most demanding is theresult for small g , namely: N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 3
Theorem 1.1. N g,n is of general type in the special cases g , and n min ( g ) n n max ( g ) with n min ( g ) , n max ( g ) given in the following table: g n min n max
10 14 18 21 25 28 32 35 38 42 46 49 52 56 60 63 66 70 74For large g we will show the following: Theorem 1.2.
The moduli space N g,n is of general type for g > . The crucial reason for Theorem 1.2 is that M g is of general type for g >
24. Thetheorem could automatically be improved if there were proof of M g being of general typefor some g <
24. Then N g,n also is of general type, by an identical proof. We emphasizethat in this case our result is uniform in the number n of nodes. There is no finite upperbound n max ( g ). The proof (presented in Section 2) uses arguments of fairly general natureand does not require elaborate calculations with carefully selected divisors. This changesdrastically in low genus g
23. It is clear from the above diagram that N g,n can only beof general type if M g, n is. In our opinion it is easier to show that M g, n is of general typethan to establish this property for N g,n . Thus we have not even attempted to considerthose values of g, n where M g, n is not known to be of general type. Thus, for low genus,we have considered individually the cases in which M g, n is known to be of general type(see [L] and [F1]). For the convenience of the reader we shall recall: M g, n is of general type for 5 g , and n > n min ( g ) , with n min ( g ) being given inthe following table: g n min N g,n in Theorem 1.1, our approach involvesderiving sufficient conditions for the property of being of general type in terms of divisorsinvariant under the group action. Basic building blocks are the Brill-Noether divisors on M g and M g + n , appropriately pulled back, and divisors of Weierstrass-type (see Section 4).For g + 1 or g + n + 1 prime, the Brill-Noether divisors have to be replaced by less efficientdivisors which we take from [EH4]. This gives, in any case, a system of linear inequalitieswhich is solvable for certain values of ( g, n ). This solvability is a sufficient condition for N g,n being of general type. In spirit our analysis is related to [FV1, FV2, FV3], but thedetailed analysis is different and requires new ideas. The explicit solution of the relevantsystems of inequalities is best done using computer algebra, paying special attention to anumber of limiting cases where the appropriate divisors have to be chosen with care.We further remark that, in the table of Theorem 1.1, the upper cut-off at n max ( g ) isthe largest integer smaller than (or in some cases equal to) ( g − −
3. and that, usingthis divisor based computational approach to prove general type in high genus g >
24 -avoiding the completely different arguments in the proof of Theorem 1.2 - would give amuch weaker result than Theorem 1.2, since one only gets general type for n bounded bythe same n max ( g ). This appearance of an upper bound for n in Theorem 1.1 is the maindifference to both our Theorem 1.2 and the results in the low-genus table for M g,n . Weremark that in order to improve our result on the upper cut-off with the techniques ofthis paper, one needs a new type of effective divisor A on M g, n of the form (using our N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 4 notation established in Section 4) aλ − b irr δ irr − X i,S b i,S δ i,S , with all coefficients nonnegative and b ,S = 0 for | S | = 2, or of the form − aλ + cψ − b irr δ irr − X i,S b i,S δ i,S , with all coefficients nonnegative and b ,S > c for | S | = 2. We do not know of any suchdivisor. In any case, at present it is not clear if our result is sharp. To clarify this questionis, in our opinion, the most interesting point left open by the results of our present paper.The outline of our paper is as follows. In Section 2 we prove Theorem 1.2. In Section 3we give sufficient conditions for our statement on N g,n being of general type in terms ofan appropriate decomposition of the canonical divisor K N g,n . In Section 4 we introducemost of the divisors needed in the subsequent analysis (i.e. all the standard divisors usedin the next section), while in Section 5 we derive most of the table (the standard part ofit) of Theorem 1.1 by solving the ensuing system of inequalities. This proof is the leasttechnical. It leaves open a number of special cases (precisely 9), each of which requiresadditional special divisors and a lot of special attention. This is the content of Section 6and concludes the proof of Theorem 1.1.Finally we remark that, though not being strictly necessary, it is convenient to employin Sections 5-6 a standard computer algebra program, e.g. Mathematica or Maple.2. Proof of Theorem 1.2
We shall prove the assertion by representing N g,n via a base and a general fibre. Recallthat for any surjective proper morphism of normal projective varieties φ : Y → X withgeneral fibre F the Kodaira dimension satisfies κ ( Y ) > κ ( X ) + κ ( F ). In particular, if F and X are of general type, then Y is of general type.Now consider the projection φ : N g,n = M g, n /G → M g , with general fibre F ≃ C n /G ≃ Sym n (Sym ( C )) = ( C / Z ) n /S n , where Sym m ( C ) := C m /S m denotes the m -fold symmetric product of the curve, and thecrepant resolution given by the Hilbert scheme (see e.g. [HL])Hilb n ( S ) → Sym n (Sym ( C )) , S = Sym ( C ) . This is of general type if S is of general type. It is well known that Sym d ( C ) is of generaltype for d g −
1. Combining these results we get that both the base M g and the generalfibre F are of general type, completing the proof. For the sake of the reader we recall theargument that Sym d ( C ) is of general type for d g − C be an irreducible smooth genus g curve, we consider for d g − π : C d → C d := Sym d ( C ) , which induces a canonical map D
7→ L D from divisors on C to divisors on C d . Denotingby K the canonical divisor on C and by ∆ / π , Prop. 2.6 of[K] gives the canonical divisor on C d as K C d = L K − ∆ / . (2.1) N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 5
Using the standard interpretation of points in C d as effective divisors on C of degree d and fixing some effective divisor D of degree g − − d gives a map α D : C d → C g − , A A + D. Composing α D with the Abel-Jacobi map u : C g − → J g − ( C ) , and taking θ to be the θ -divisor on the Jacobi torus J g − ( C ), Prop. 2.3 and Prop. 2.7.of [K] give u ∗ α ∗ D θ = L K − ∆ / − L D . Combining this with (2.1) yields K C d = u ∗ α ∗ D θ + L D , representing the canonical divisor of C n as a sum of an ample divisor (because θ is ample)and an effective divisor. This is a well known criterion for C d to be of general type (seee.g. our Section 3 below).3. Moduli spaces of nodal curves
The aim of this section is to develop a sufficient condition for N g,n being of generaltype. This requires a basic understanding of the Picard group Pic( N g,n ) and an explicitdescription of the boundary divisors and tautological classes on N g,n which we shall al-ways consider as G -invariant divisors on M g, n (any such divisor descends to a divisoron N g,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 Picard group on the moduli stackwith that of the corresponding coarse moduli space.In particular, we recall the notion of the Hodge class λ on M g,n , which automaticallyis G − invariant and thus gives the Hodge class λ on N g,n (where, by the usual abuse ofnotation, we denote both classes by the same symbol).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 φ : M g,n → M g .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 given N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 6 by 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 , (3.1)and introduce ψ = P ni =1 ψ i . Clearly, on M g, n , the class ψ is G − invariant.We shall now consider the action of G on these divisor classes on M g, n . Clearly, λ, ψ and δ irr are G − invariant. Furthermore, δ i,S is mapped into δ j,T by some element of G ifand only if i = j, | S | = | T | and S and T contain the same number of pairs. This impliesthat one gets a G − invariant divisor class by setting δ i ; a,b = X S δ i,S , where the sum is taken over all subsets S containing precisely a pairs and b single points.As a first step in the direction of our sufficient criterion we need the following result onthe geometry of the moduli space N g,n . Theorem 3.1.
The moduli space N g,n has only canonical singularities or in other words:The singularities of N g,n do not impose adjunction conditions , i.e. if ρ : ˜ N g,n → N g,n isa resolution of singularities, then for any ℓ ∈ N there is an isomorphism ρ ∗ : H (( N g,n ) reg , K ⊗ ℓ ( N g,n ) reg ) → H ( ˜ N g,n , K ⊗ ℓ ˜ N g,n ) . (3.2) Here ( N g,n ) reg denotes the set of regular points of N g,n , considered as a projective variety,and K ˜ N g,n , K ( N g,n ) reg denote the canonical classes on ˜ N g,n and ( N g,n ) 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 covers our case.Theorem 3.1 implies that the Kodaira dimension of N g,n equals the Kodaira-Iitakadimension of the canonical class K N g,n . In particular, N g,n is of general type if K N g,n isa positive linear combination of an ample and an effective rational class on N g,n . It isconvenient to slightly reformulate this result. We need Proposition 3.2.
The class ψ on M g, n is the pull-back of a divisor class on N g,n whichis big and nef.Proof. Farkas and Verra have proven in Proposition 1.2 of [FV2] that the S n -invariantclass ψ descends to a big and nef divisor class N g, n on the quotient space M g, n /S n . N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 7
Consider the sequence of natural projections M g, n π −→ M g, n /G ν −→ M g, n /S n . Then ν ∗ ( N g, n ) is a big and nef divisor on N g,n = 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 is precisely δ , . In fact a quotient map X → X/G is ramified in a point x ∈ X exactly if thereexists a non-trivial element g ∈ G such that g ( x ) = x . In our case an element g ∈ G acts on a class of pointed curves by g [ C, x , . . . , x n ] = [ C ; x g (1) , . . . , x g (2 n ) ]. By standardresults the pointed curves ( C, x , . . . , x n ) and ( C ; x σ (1) , . . . , x σ (2 n ) ) are isomorphic for some σ = id ∈ S n , if and only if C has a rational component with exactly two marked pointsand σ is the transposition switching these two points. Since our group G contains exactlythose transpositions belonging to the pairs ( x i , y i ), the ramification divisor is δ , .Furthermore, standard results on the pullback give K := π ∗ ( K N g,n ) = K M g, n − ram π = 13 λ + ψ − δ − δ , ∅ − δ , . (3.3)Here δ is the class of the boundary of M g, n , i.e. δ = δ irr + X i ⌊ g ⌋ X S ⊂{ ,..., n } δ i,S . 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 N g,n is of general type.4. Effective G -invariant divisors on M g, n In this section we construct G − invariant effective divisors on M g, n . First we recallthe following standard result.
Proposition 4.1.
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 . The most natural effective divisors on M g are the Brill-Noether divisors parametrizingall curves C possessing a g rd for fixed g, r, d with Brill-Noether number ρ ( g, r, d ) = − g + 1 is composite. We recall from [EH4]: Lemma 4.2.
Assume that g + 1 is not prime and fix some integers r, s > such that g + 1 = ( r + 1)( s − . Then BN g := { [ C ] ∈ M g | C carries a g rrs − } is an (effective) divisor on M g . Furthermore the class of its compactification as a Q -divisor is given by [ BN g ] = c (cid:16) ( g + 3) λ − g + 16 δ − ⌊ g ⌋ X i =1 i ( g − i ) δ i (cid:17) , for some positive rational number c . Note that only the constant c depends on the choice of r and s . Note also that g + 1 =( r +1)( s −
1) implies that the Brill-Noether number satisfies ρ ( g, r, rs −
1) = −
1. Therefore BN g parametrises precisely those (non-general) curves violating the condition of the Brill-Noether Theorem (see [GH]), which states that a general curve carries a g rd , if and only if ρ ( g, r, d ) > N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 8
Now consider the forgetful map φ : M g, n → M g and recall from the introduction themap χ : M g, n → M g + n .If neither g + 1 nor g + n + 1 are prime, we pull back the class of [ BN g ] as a Q -Divisor(up to multiplication with a positive scalar) and set B = 1 c φ ∗ [ BN g ] , D = 1 c χ ∗ [ BN g + n ] . (4.1)We shall now apply results of [AC2], where the pull-backs of the divisor classes ψ i , δ irr , δ i,S are computed under the forgetful map forgetting one point and under the gluing map glu-ing two marked points. Thus repeatedly applying Lemma 1.2 and Lemma 1.3 of [AC2]yields Lemma 4.3.
With the above notation, one has B = b λ λ + 0 · ψ − b δ irr − ⌊ g ⌋ X i =1 X S b i δ i,S (4.2) with b λ = g + 3 , b = g + 16 , b i = i ( g − i ) . Furthermore D = d λ λ + d ( ψ − δ irr − ⌊ g ⌋ X i =0 X a + b n,b =0 δ i ; a,b ) − ⌊ g ⌋ X i =0 n X a =0 d i + a δ i ; a, (4.3) with d λ = g + n + 3 , d = g + n + 16 , d i = i ( g + n − i ) . In particular, both divisors are G − invariant. The problem is that these divisors willnot exist for g + 1 or g + n + 1 prime. In that case we have to use less efficient divisorsparametrizing curves that violate the Gieseker-Petri condition.We recall that a curve C satisfies the Gieseker-Petri condition, if for every line bundle L the natural map µ : H ( C, L ) ⊗ H ( C, K ⊗ L − ) → H ( C, K )is injective.We also recall from [EH4] Theorem 2 the following divisor.
Lemma 4.4.
Assume that g = 2 d − is even, then GP g := { [ C ] ∈ M g | C violates the Gieseker-Petri condition } is an (effective) divisor on M g . Furthermore the class of its compactification is given by [ GP g ] = c ( a λ λ − g X i =0 a i δ i ) , for some rational number c > and a λ = 6 d + d − , a = d ( d − , a = (2 d − d − , a i +1 > a i . N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 9
Similar to (4.1), we pull back [ GP ] along the forgetful map φ and the gluing map χ and set E = 1 c φ ∗ [ GP g ] , F = 1 c χ ∗ [ GP g + n ] . (4.4)By repeatedly applying [AC2], Lemma 1.2 and Lemma 1.3, we obtain E = e λ λ + 0 · ψ − e δ irr − g X i =1 X S e i δ i,S , (4.5)with e λ = 6( g + g − , e = ( g g , e = ( g − g , e i +1 > b i , and F = f λ λ + f ( ψ − δ irr − ⌊ g ⌋ X i =0 X a + b n,b =0 δ i ; a,b ) − ⌊ g ⌋ X i =0 n X a =0 f i + a δ i ; a, , (4.6)with f λ = 6( g + n + g + n − , f = ( g + n g + n , f = ( g + n − g + 3 n ,f i +1 > ˜ b i . Now observe that the divisor classes B and E are effective by Proposition 4.1, becausethe forgetful map φ is onto. The divisor class F is effective because the general nodalcurve is Gieseker-Petri general. This is already implicitly contained in [G] (see also [EH2]),in the proof of the Gieseker-Petri Theorem. Since the proof uses deformation to a nodalcurve, it actually shows the above statement. Effectiveness of D follows in the same wayobserving that the Gieseker-Petri condition implies the Brill-Noether condition. Alterna-tively, one could consider the proof of the Brill-Noether Theorem in [GH], which also usesdeformation to a nodal curve.Finally we need divisors of Weierstrass-type. We recall from [L], Section 5, the divi-sors W ( g ; a , . . . , a m ) on M g,m , where a i > P a i = g . They are given by thelocus of curves C with marked points p , . . . , p m such that there exists a g g on C con-taining P i m a i p i . We want to minimize the distance between the weights a i . Thus wedecompose g = km + r , with r < m , and set W m = W ( g ; a , . . . , a m ) , a j = k + 1 (1 j r ) , a j = k ( r + 1 j m ) . (4.7)This gives, in view of [L], Theorem 5.4, W 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 , (4.8)where higher order means a positive linear combination of δ i,S where either i > | S | > W m we want to generate a G − invariant divisor class W on M g, n , by summingover 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 φ S,T : M g, n → M g,m (4.9) N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 10 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 a ( g, n, m, r ) := (cid:18) nr (cid:19)(cid:18) n − rm − r (cid:19) such projections. With this notation we introduce W := X S,T φ ∗ S,T W m = − w λ λ + w ψ ψ + 0 · δ irr − X s > w s X | S | = s δ ,S − higher order boundary terms , (4.10)where higher order denotes a positive linear combination of boundary divisors δ i,S with i > w s > sw ψ > w ψ for s > , (4.11) w λ = a ( g, n, m, r ) = − (cid:18) nr (cid:19)(cid:18) n − rm − r (cid:19) , (4.12) 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 , (4.13) 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 . (4.14)Equation (4.13) is proved by applying pullback to (4.8), using ω := P ni =1 ω i , X S,T r X i =1 π ∗ S,T ω i = a ( g, n, m, r ) r nω = (cid:18) n − r − (cid:19)(cid:18) n − rm − r (cid:19) ω (4.15)and X S,T m X i = r +1 π ∗ S,T ω i = a ( n, m, g ) m − r n ω = (cid:18) n − r (cid:19)(cid:18) n − r − m − r − (cid:19) , (4.16)noting that equation (3.1) implies ω = ψ − X S | S | δ ,S . (4.17)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 (4.14) and the bound in (4.11) are generated by the change ofbasis given in (3.1).It turns out that we will be able to improve the upper bound on n given in Proposition5.1 by replacing W with divisors parametrizing curves, that fail the so called MinimalResolution conjecture , see [F1], Theorem 4.2.
N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 11
Lemma 4.5.
Fix integers g, r > and k g and set m = (2 r + 1)( g − − k . Then Mrc rg,k := { [ C, x , . . . , x m ] ∈ M g,m | h ( C, ∧ i M K C ⊗ K ⊗ ( r +1) C ⊗ O C ( − x − · · · − x m )) > } is a divisor on M g,m . Furthermore the class of its compactification is given by [ Mrc rg,k ] = 1 g − (cid:18) g − k (cid:19) ( − a λ λ + a ψ ψ + a irr δ irr − X i,s a i,s X | S | = s δ i,S ) (4.18) where a λ = 1 g − (cid:16) ( g − g − r + 6 r + r ) + k (24 r + 10 k + 10 − g − rg ) (cid:17) ,a ψ = rg + g − k − r − ,a irr = 1 g − (cid:16)(cid:18) r + 12 (cid:19) ( g − g −
2) + k ( k + 1 + 2 r − rg − g ) (cid:1) ,a ,s = (cid:18) s + 12 (cid:19) ( g −
1) + s ( rg − r − k ) and a i,s > a ,s . (4.19)Note that [ Mrc rg,k ] is already G -invariant.For g odd and 2 n > g − r, k such that 2 n = m = (2 r + 1)( g − − k . In fact we will only consider 2 n > g + 1 and choose k g −
2. This will makesome computations easier and uniquely determine r, k for given g, n . So for g odd andsuch r, k we define the divisor U as the class of Mrc rg,k as a Q -divisor (up to multiplicationwith a positive scalar)by U := [ Mrc rg,k ] = − u λ + u ψ ψ + u irr δ irr − X i,s u i,s X | S | = s δ i,S (4.20)with u λ , u ψ , u irr , u i,s being a λ , a ψ , a irr , a i,s from (4.19).For g even we set 2 n − m = (2 r + 1)( g − − k and pull back via all possibleforgetful maps φ i : M g, n → M g,m , forgetting the i th point. Again we only consider k g − m > g + 1. We emphasize that, just as above for U , theintegers r and k are uniquely determined by g and n . This gives V := n X i =1 φ ∗ i [ Mrc rg,k ] = − v λ + v ψ ψ + v irr δ irr − X i,s v i,s X | S | = s δ i,S (4.21)with v λ = 2 ng − (cid:16) ( g − g − r + 6 r + r ) + k (24 r + 10 k + 10 − g − rg ) (cid:17) ,v ψ =(2 n − rg + g − k − r − ,v irr = 2 ng − (cid:16)(cid:18) r + 12 (cid:19) ( g − g −
2) + k ( k + 1 + 2 r − rg − g ) (cid:1) ,v , =2( rg + g − k − r −
1) + (2 n − g − rg − r − k ) v i,s > v , . (4.22) N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 12 Standard cases in the proof of Theorem 1.1: Reduction to a systemof inequalities
As mentioned in Section 3, the moduli space N g,n is of general type, if and only if itscanonical divisor K N g,n is the sum of an ample divisor and effective divisors. We showthis by decomposing K = π ∗ ( K N g,n ) on M g, n as a sum of a positive multiple of ψ , ofnon-negative multiples of the G -invariant divisors constructed in the last section and ofnon-negative multiples of λ, δ irr and the δ i ; a,b .We will begin by using the divisor class W to show: Proposition 5.1. N g,n is of general type in the special cases g , and n min ( g ) n n max ( g ) with n min ( g ) , n max ( g ) given in the following table: g n min n max
10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42Then we will use the divisor classes U and V to improve the upper bound n max andextend our result to g = 5 , Proposition 5.2. N g,n is of general type in the special cases g , and n min ( g ) n n max ( g ) with n min ( g ) , n max ( g ) given in the following table: g n min n max
10 14 18 21 25 28 32 35 38 42 46 49 52 56 60 63 66 70 74Note that the lower bound n min differs from Theorem 1.1 only in the cases g = 5 , , g = 5 and g = 6 this is due to the fact that the preliminary upper bound actually liesbelow the lower bound. Proof of Proposition 5.1.
For g + 1 and g + n + 1 both composite, we will search for anon-negative linear combination of the divisors B, D and W such that K − xB − yD − zW (possibly up to an arbitrarily small multiple ǫψ , see equations (5.7), (5.8) below and ourdiscussion of Case II) becomes an effective combination of the tautological and boundaryclasses. (Whenever g + 1 is prime we replace B by E and whenever g + n + 1 is prime wereplace D by F .) This will translate into a system of inequalities, one for each tautologicalor boundary class. However it is easy to check that only the inequalities imposed by λ, ψ, δ irr , δ and δ are relevant. The others are automatically satisfied as soon asthese 5 are.The table of relevant coefficients is: λ ψ δ irr δ , δ , B b λ − b D d λ d − d − d − d E e λ − e F f λ f − f − f − f W − w λ w ψ − w − w K
13 1 − − − N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 13 with b λ = g + 3 , b = g + 16 ,d λ = g + 3 , d = g + 16 , d = g + n − ,e λ = 6( g + g − , e = ( g g ,f λ = 6( g + n + g + n − , f = ( g + n g + n , f = ( g + n − g + 3 n , and w λ , w ψ , w from (4.12), (4.13) and (4.14).When looking at the divisor W defined in equation (4.10) we have the condition m min(2 n, g ). For the remainder of the paper we shall always set m = min(2 n, g ) and g = mk + r with r < m . We shall treat separately the (easy)Case I: 2 n < g (which we shall split into 2 n g − n = g − g n .Looking at the coefficients w ψ and w of this divisor (see (4.13) and(4.14)) it is easy toshow that w > w ψ for 2 n g − w = 3 w ψ for 2 n = g − n = g and w < w ψ for 2 n > g . This motivates treating 2 n = g − n g − g + 1 composite, we have 3 w ψ < w and therefore the optimaldecomposition is x = b , y = 0 , z = w . This is exactly the same decomposition Loganuses to prove M g,n being of general type. It satisfies the inequalities imposed by δ irr and δ as equalities and also satisfies the inequalities imposed by ψ and δ , . Only theinequality imposed by λ remains to be checked:( λ ) : 2 b λ b − w λ w
13 (5.2)Whenever this is true for a given g and n , we get the decomposition K = 2 b B + 3 w W + (1 − w ψ w ) ψ + ”effective” . (5.3)For 2 n g − g + 1 prime (and thus g > B by E in (5.2). Thus,whenever we have ( λ ) : 2 e λ e − w λ w , (5.4)we actually get the desired decomposition K = 2 e E + 3 w W + (1 − w ψ w ) ψ + ”effective” , (5.5)proving that N g,n is of general type. Checking the inequality (5.4) by use of a smallcomputer program establishes the table for 2 n g − n = g − w ψ = w ,giving a vanishing coefficient of ψ . We therefore need to use the divisor D for g + n + 1composite and F for g + n + 1 prime. Note that g + 1 = 2 n + 2 is composite. Whenever( λ ) : 2 b λ b − w λ w <
13 (5.6)
N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 14 we can use the decomposition K = 2 b B + 2 ǫd D + 1 − ǫw ψ W + ǫψ + ”effective” (5.7)or K = 2 b B + 3 ǫf F + 1 − ǫw ψ W + ǫψ + ”effective” (5.8)for some ǫ > n > g . Let us begin by assuming both g + 1and g + n + 1 are composite. We start similarly to Case I, by trying to decompose thecanonical class as K = xB + yD + zW + ǫψ + ”effective”for some non-negative x, y, z and positive ǫ .The coefficients of W become especially easy: w λ = (cid:18) ng (cid:19) , w ψ = (cid:18) n − g − (cid:19) , w = 2 (cid:18) n − g − (cid:19) + (cid:18) n. − g − (cid:19) . We shall first show that there is a finite upper bound for n , i.e. for fixed g and n sufficiently large, the conditions imposed by ψ, δ , and δ , cannot be satisfied simul-taneously. In fact, let us consider the corresponding system of 3 linear inequalities for thevariables y, z , read off the table (5.1):( ψ ) : d y + w ψ z < δ , ) : − d y − w z − δ , ) : − d y − w z − n > g ) any solution y, z of (5.9) automatically is positive:( ψ ) + ( δ , ) implies z > ψ ) + ( δ , ) then gives y > . Then the inequality( ψ ) + ( δ , ) gives z > w − w ψ > w ψ (5.10)and inequality d ( ψ ) + d ( δ , ) gives z < d − d d w ψ − d w . (5.11)This is solvable if and only if( d − d )( w − w ψ ) − ( d w ψ − d w ) > . (5.12)Consider p g ( n ) := (12 n − (cid:18) n − g − (cid:19) − (cid:16) ( d − d )( w − w ψ ) − ( d w ψ − d w ) (cid:17) = − n + (2 g − n + g − g + 9 (5.13)as a polynomial in n . Then p g ( n ) > g − − p − g + 36 g ) < n <
14 (2 g − p − g + 36 g ) . (5.14)As we are only interested in solutions with n ∈ N , we get N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 15 n
14 (2 g − p − g + 36 g ) = 2 g − . (5.15)This establishes the existence of an upper bound for n , as claimed above.Now let us assume, that g n g − y, z ) is a solution of the threeinequalities (5.9) (preferably with z as large as possible). We set x = 1 b (2 − d y )and show that ( x, y, z ) satisfies also the two remaining inequalities:( λ ) : b λ x + d λ y − w λ z δ irr ) : − b x − d y − δ irr ) will be satisfied as equality, and we are left with checking ( λ ).If g + 1 is prime we replace B by E and for g + n + 1 prime we replace D by E . Asthere are only finitely many cases to check, we have done this by explicitly computing thedecomposition with the help of a simple program in Mathematica. This proves all theresults tabled in Proposition 5.1. (cid:3) In order to show Proposition 5.2 we replace the divisor W by either U (for g odd) orby V (for g even). Proof of Proposition 5.2.
As a general remark, note that the divisors
U, V do not exist if n is too small, depending on g , i.e. for 2 n < g −
4. In this case, we simply use Proposition5.1 to generate the table in Proposition 5.2.In all other cases we want to decompose K as a sum of xB or xE , yD or yF , zU or zV with non-negative x, y, z , some positive multiple ǫψ of the point bundles and an effectivecombination of the tautological classes and boundary divisors. So let us consider the tableof relevant coefficients λ ψ δ irr δ , δ , B b λ − b D d λ d − d − d − d E e λ − e F f λ f − f − f − f U − u λ u ψ u irr − u , − u , V − v λ v ψ v irr − v , − v , K
13 1 − − − b λ , b are given in (4.2), d λ , d , d in (4.3), e λ , e in (4.5), f λ , f , f in (4.6), u λ , u ψ , u irr , u , in (4.20) and v λ , v ψ , v irr , v , in (4.21). λ, ψ, δ irr , δ , , d , each determine a linear inequality that can be read off from thetable. Analogous to (5.9) we begin by considering the system of inequalities( ψ ) : d y + u ψ z < δ , ) : − d y − u , z − δ , ) : − d y − u , z − . (5.18)This corresponds to the case g + 1 composite, g + n + 1 composite and g odd. It is oneof 6 possible cases and will be the only one which we shall discuss in detail. N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 16
By Gaussian elimination (compare (5.12)) this is solvable, if and only if0 p ( g, n, r, k ) := 6 (cid:16) ( d − d )( u , − u ψ ) − ( d u ψ − d u , ) (cid:17) = 9 − g + 3 g + k + gk − n + 3 gn + kn + r − g r + nr − gnr (5.19)This inequality defines the upper cut-off in the table of Proposition 5.2 (in the specialcase considered here), which improves the table of Proposition 5.1. We the have to showthat for all these values of g and n we actually get a solution of our system of inequalities.This we did by computing, in all these cases, an explicit solution using Mathematica.All other cases are treated in a similar way: For g even, the divisor U has to be replacedby V , if g + n + 1 is prime, one has to replace D by F , and if g + 1 is prime, B neddsto by replaced by E . In each of these cases one gets a corresponding system of linearinequalities and a neccessary upper cut-off for n . This upper bound defines n max ( g ) in ourtable. As above, we have then computed a solution of the ensuing system of inequalitiesby Mathematica, for all relevant values of g and n . This gives the improved table inProposition 5.2. (cid:3) Proof of Theorem 1.1: Special computations
The aim of this section is to prove Theorem 1.1. In view of Propostion 5.2 this boilsdown to specific calculations for each of the values of ( g, n ) contained in the table ofTheorem 1.1, but not in the table of Proposition 5.2. In each case, we shall need addi-tional divisors, specifically adapted to the case at hand. Our standard references for thesedivisors are [L], [F1]. N , and N , We use the divisor class Z , on M from [F1], Theorem 1.4. This is a divisor thatviolates the slope conjecture, i.e. it has a smaller slope than the Brill-Noether divisor.We have s ( Z , ) = 7, i.e Z , = c (7 λ − δ irr − P i a i δ i ) on M . Take Z = φ ∗ ( Z , ), thepull-back via the forgetful map φ : M , n → M . Then for n = 6 we can decompose K as an effective combination of Z, F, W , some positive multiple of ψ and an effectivecombination of the tautological and boundary classes. For n = 6 we can decompose K asan effective combination of Z, D, W , some positive multiple of ψ and an effective combi-nation of the tautological and boundary classes. N , We use the divisor class Z , on M from [F1] Theorem 1.4. This divisor has slope s ( Z , ) = , which is smaller than the slope of the Brill-Noether divisor. We set Z = φ ∗ ( Z , ) the pull-back via the forgetful map φ : M , → M . The canonicaldivisor will decompose as an effective combination of Z, W some positive multiple of ψ and an effective combination of the tautological and boundary classes. N , Similarly to the above divisors, we use the divisor class Z , on M from [F1], Corollary1.3. This divisor has slope s ( Z , ) = , which once again is smaller than the slopeof the Brill-Noether divisor. We set Z = φ ∗ ( Z , ), the pull-back via the forgetful map φ : M , → M . The canonical divisor will decompose as an effective combination of Z, W , some positive multiple of ψ and an effective combination of the tautological and N THE KODAIRA DIMENSION OF THE MODULI SPACE OF NODAL CURVES 17 boundary classes. N , Here we use the divisor class D = 13245 λ − δ irr − δ − . . . on M from [FV4]and pull it back along the forgetful map φ : M , → M . The canonical divisor willdecompose as an effective combination of φ ∗ ( D ) , F, W some positive multiple of ψ and aneffective combination of the tautological and boundary classes. N , and N , For g = 22 there are no Brill-Noether divisors, because 22 + 1 is prime. Instead of usingthe divisor class E we take a Brill-Noether divisor B = c (26 λ − δ irr − δ − . . . ) on M .We pull this divisor back along χ : M , → M and then along all possible forgetfulmaps φ S : M , → M , or M , → M , .On M , , adding all these pull-backs, gives us the divisor class L , = 12 c (13 λ + ψ − δ irr − P | S | =2 δ ,S − . . . ). This divisor class alone proves the canonical divisor K to beeffective. By using a linear combination with ǫ E, ǫ W for some suitable ǫ , ǫ > K to be big.On M , , adding all these pull-backs, gives us the divisor class L , = 30 c (13 λ + ψ − δ irr − P | S | =2 δ ,S − . . . ). Using L , and W we can show K to be big. N , We can show that N , is of general type by using the divisor classes B and Nfold , = c (35 λ + 54 ψ − δ irr − P | S | =2 δ ,S − . . . ) from [F1], Theorem 4.9. N , We take the divisor class
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Humboldt Universit¨at Berlin, Institut f¨ur Mathematik, Rudower Chausee 25, 12489Berlin, Germany
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