ON THE EFFECTIVE CONE OF M g,n SCOTT MULLANE 7 Abstract. For every g≥2 and n≥g+1 we exhibit infinitely many extremal effective divisors in 1 Mg,n coming from the strata of abelian differentials. 0 2 Contents n a 1. Introduction 1 J 2. Preliminaries 3 0 2 3. Divisors from the strata of abelian differentials 8 4. Moving curves 10 ] G 5. Covering curves and the effective cone of Mg,n 11 References 15 A . h t a m 1. Introduction [ The space of holomorphic differentials or one-forms on a curve provides the canonical map of 1 the curve into projective space. Hence the differentials encode a wealth of information about a v curve. Instead of considering all differentials on a fixed curve, we can extract the information held 3 by differentials from a different perspective. By fixing the type of differential and also allowing the 9 8 curve to vary, we obtain the moduli space of abelian differentials H(κ) consisting of pairs (C,ω) 5 where ω is a holomorphic or meromorphic differential on a smooth curve C and the multiplicity of 0 the zeros and poles of ω is fixed of type κ, an integer partition of 2g−2. Equivalently, a flat metric . 1 on C with a finite number of singularities gives a translation atlas on the curve away from these 0 singularities. Hence a pair (C,ω) is also known as a translation surface and can be represented 7 1 as polygons in the plane with parallel side identifications. Previous seminal work exposes the : fundamental algebraic attributes of these spaces [Mc][KZ][EMa][EMM][Fil]. v i Subvarieties of codimension one are the vanishing locus of a global section of a line bundle and X are known as effective divisors. Geometrically defined effective divisors of Brill-Noether and Petri r a type were first used by [HMu][H][EH] to show that Mg is of general type for g > 23. This was extended by Koszul divisors to g ≥ 22 by [F1][F2]. Similar results were obtained by [L] for M g,n with n>0, which were again sharpened by [F2]. For any fixed holomorphic signature κ, the locus of curves admitting a one-form of type κ forms a subvariety in M . In [M1] the author presents a closed formula for the divisor class in M g g of the closure of all possible codimension one loci of this type. This generalised the two known cases of the loci of curves with an extremal Weierstrass point [D] and the loci of curves with a subcanonical pencil [T]. In M , effective divisors from the strata of abelian differentials have g,n been previously used under different guises in a number of places, for example, divisors of Brill- Noether type [L], the pullback of the theta divisor [Mu¨][GZ], the loci of points in the support of Date: January 23, 2017. 1 an odd theta characteristic [F3] and the anti-ramification locus [FV2]. In [M2] the author uses the degeneration of abelian differentials [BCGGM] and maps between moduli spaces to compute many new classes of divisors of this type as well as efficiently reproducing these previously known classes. The stratum of canonical divisors with signature κ=(k ,...,k ) is a subvariety of M , 1 m g,m P(κ)∶={[C,p ,...,p ]∈M ∣ k p +...+k p ∼K }. 1 m g,m 1 1 m m C The moduli space of twisted canonical divisors of type κ, a closure of this variety in M , was g,m providedby[FP]whoshowedthatthisspacecontainedthemaincomponentcomingfromcanonical divisors on smooth curves and extra components completely contained in the boundary of M . g,m The global residue condition that distinguishes the main component was given by [BCGGM], pro- viding a full compactification of P(κ). Forgetting points to obtain a codimension one subvariety and taking the closure we obtain the divisor Dn ={[C,p ,...,p ]∈M ∣ [C,p ,...,p ]∈M with ∑k p ∼K }, κ 1 n g,n 1 m g,m i i C in M for κ = (k ,...,k ) with ∑k = 2g −2 where m = n+g −2 or n+g −1 for holomorphic g,n 1 m i and meromorphic signature κ respectively. For g ≥ 2 this divisor is irreducible unless all k are i even where there are two irreducible components based on spin structure that we denote by the indices odd and even. In §3 we review the known classes Dn for meromorphic signature κ due κ to [Mu¨][GZ][M2]. Tensoring global sections of any two line bundles L ,L we obtain a global section of the tensor 1 2 product L ⊗L . Hence the sum of two effective divisors on a projective variety X is an effective 1 2 divisor and there is a natural cone structure on effective divisors on X. Taking the closure of this cone we obtain Eff(X), the pseudo-effective cone of X. A fundamental problem in the birational geometry of moduli spaces is to understand the structure of this cone. The complete structure of the pseudo-effective cone for M remains an open problem. [CTel1] found finitely many extremal g,n effective divisors in M for every n ≥ 7 indexed by irreducible hypertrees. [CC] gave infinitely 0,n manyextremaleffectivedivisorsinM foreveryn≥3, showinginthesecasesthepseudo-effective 1,n cone cannot be rational polyhedral. In general genus g ≥ 2, [FV1][FV2] gave a finite number of extremal divisors for any fixed g and n with g−2≤n≤g. A moving curve B in a projective variety X is a curve class such that irreducible curves with numericalclassequaltoB coveraZariskidensesubsetofX. HenceB⋅D≥0foranypseudo-effective divisor D, further, if B⋅D=0 then D lies on the boundary of the pseudo-effective cone. A covering curve B of an effective divisor D is a curve class such that irreducible curves with numerical class equal to B cover a Zariski dense subset of D. If B⋅D<0 then D is extremal in the pseudo-effective cone. Taking a fibration of P(κ) for a meromorphic signature κ we obtain curves in M , g,n m Bn ∶={[C,p ,...,p ]∈M ∣ fixed general [C,p ,...,p ]∈M and ∑k p ∼K }. κ 1 n g,n g+2 m g,m−g−1 i i C i=1 For m=∣κ∣≥n+g these curves provide covering curves for M . In §5 we show these curves yield g,n the following information about the pseudo-effective cone. Theorem 1.1. For g ≥2, meromorphic signature κ with ∣κ∣=n+g−1 for n≥g+1, Bn ⋅Dn =0. κ,1,−1 κ Hence the irreducible components of the divisors Dn lie on the boundary of the pseudo-effective κ cone. Further, if all k are even then i aDn,odd+bDn,even κ κ for any a,b≥0, lies on the boundary of the pseudo-effective cone. 2 For any divisor Dn, curves with numerical class equal to Bn cover a Zariski dense subset of the κ κ divisor. For certain signatures we are able to show the curve Bn is a component of a specialisation κ of Bn with the other component being completely contained in the boundary of M . The κ,1,−1 g,n positive intersection of these boundary curves with Dn gives κ Bn⋅Dn <0, κ κ showing these divisors to be extremal and rigid. Theorem1.2. ThedivisorsDg+1 inM forg ≥2arerigidandextremalford=(d ,d ,d ,1g−2) d,1g−1 g,g+1 1 2 3 with d +d +d =1 and ∑ d ≤−2. Hence Eff(M ) is not rational polyhedral for g ≥2, n≥g+1. 1 2 3 di<0 i g,n A Q-factorial projective variety X is a Mori dream space if it is of the simplest type from the perspective of the minimal model program. Introduced by [HK], if the Cox ring of X is finitely generated and the Neron-Severi group of X is Pic(X)⊗Q then the nef and semi-ample cones of divisors on X correspond and the effective cone of divisors on X is rational polyhedral and has a specific chamber decomposition. By providing a nef divisor that is not semi-ample, [K] showed M is not a Mori dream space for g ≥ 3 and n ≥ 1. [CC] exhibited infinitely many extremal g,n divisors in the effective cone of M for n ≥ 3 showing that these spaces are not Mori dream 1,n spaces. M was long thought to be a Mori dream space due to its similarity to a toric variety. 0,n This was known for n ≤ 6, however, [CTel2] showed M is not a Mori dream space for n ≥ 134. 0,n This result was improved to n≥13 by [GK] and then to n≥10 by [HKL] who further showed that the intermediate open cases would require a new method. Theorem 1.2 sheds light on the open cases of genus g =2, providing the following result. Corollary 1.3. M is not a Mori dream space for n≥3. 2,n Acknowledgements. I am grateful to Dawei Chen, Joe Harris and Anand Patel for many stimu- lating discussions on topics related to this paper. 2. Preliminaries 2.1. Strata of abelian differentials. The stratum of abelian differentials with signature κ = (k ,...,k ), a non-zero integer partition of 2g−2 is defined as 1 n H(κ)∶={(C,ω) ∣ g(C)=g, (ω)=k p +...+k p , for p distinct} 1 1 n n i where ω is a meromorphic differential on C. Hence H(κ) is the space of abelian differentials with prescribed multiplicities of zeros and poles given by κ. By relative period coordinates H(κ) has dimension 2g+n−1 if κ is holomorphic (all k >0) and 2g+n−2 if κ is meromorphic (some k <0). i i The stratum of canonical divisors with signature κ is defined as P(κ)∶={[C,p ,...,p ]∈M ∣ k p +...+k p ∼K }. 1 n g,n 1 1 n n C By forgetting the ordering of the zeros or poles of the same multiplicity we obtain the projectivisa- tion of H(κ) under the C∗ action that scales the differential ω. Hence P(κ) has dimension 2g+n−2 or 2g+n−3 for holomorphic or meromorphic κ respectively. A theta characteristic η is a line bundle on a smooth curve C such that η⊗2 ∼ K . The parity C of h0(C,η) is known as the spin structure of η and is deformation invariant [Mu][A]. If all k are i even then any (C,ω) with (ω)=∑n k p specifies a theta characteristic on the underlying curve i=1 i i n k η ∼∑ ip . i 2 i=1 3 In these cases the loci H(κ) and P(κ) are reducible with disjoint components with even and odd parity of h0(C,η) known as spin structure. For specific signatures an extra component appears as differentials on a hyperelliptic curves resulting from pulling back a degree g−1 rational function under the unique hyperelliptic cover of P1. [KZ] showed that this resulted in a distinct hyperelliptic component for signatures κ=(2g−2) and (g−1,g−1) for g ≥4. Hence H(κ) and hence P(κ) has at most 3 connected components for holomorphic κ. [Bo] showed that for g ≥2 in the meromorphic case, in addition to signatures with all k even, signatures with all even positive entries and poles of the form {−1,−1} also had two i spin structure components based on the parity of the line bundle k η ∼ ∑ ip . i 2 ki>0 Further, H(κ) and hence P(κ) contain a hyperelliptic component distinct from any possible spin structure component if the zeros of κ are of the form {2n} or {n,n} for any positive integer n and the poles are of the form {−2p} or {−p,−p} for any positive integer p≥2. 2.2. Degeneration of abelian differentials. The moduli space of twisted canonical divisors of type κ=(k ,...,k ) was defined by [FP]. Consider a stable pointed curve [C,p ,...,p ]∈M . A 1 n 1 n g,n twisted canonical divisor of type κ is a collection of (possibly meromorphic) divisors D on each j irreducible component C of C such that j (a) The support of D is contained in the set of marked points and the nodes lying in C , j j moreover if p ∈C then ord (D )=k . i j pi j i (b) If q is a node of C and q ∈C ∩C then ord (D )+ord (D )=−2. i j q i q j (c) Ifq isanodeofC andq ∈C ∩C suchthatord (D )=ord (D )=−1thenforanyq′ ∈C ∩C i j q i q j i j we have ord (D )=ord (D )=−1. We write C ∼C . q′ i q′ j i j (d) If q is a node of C and q ∈C ∩C such that ord (D )>ord (D ) then for any q′ ∈C ∩C i j q i q j i j we have ord (D )>ord (D ). We write C ≻C . q′ i q′ j i j (e) There does not exist a directed loop C ⪰C ⪰...⪰C ⪰C unless all ⪰ are ∼. 1 2 k 1 In addition to the main component containing canonical divisors of type κ on smooth pointed curves, [FP] showed this space included extra components completely contained in the boundary. The global residue condition, recently provided by [BCGGM], distinguishes the main component from the boundary components giving a full compactification for the strata of abelian differentials. Let Γ be the dual graph of C. A twisted canonical divisor of type κ is the limit of twisted canonical divisors on smooth curves if there exists a collection of meromorphic differentials ω on C with i i (ω )=D that satisfy the following conditions i i (a) If q is a node of C and q ∈ C ∩C such that ord (D ) = ord (D ) = −1 then res (ω )+ i j q i q j q i res (ω )=0. q j (b) There exists a full order on the dual graph Γ, written as a level graph Γ, agreeing with the order of ∼ and ≻, such that for any level L and any connected component Y of Γ that >L does not contain a prescribed pole we have ∑ res (ω )=0 q i level(q)=L, q ∈C ∈Y i Part (b) is known as the global residue condition. We provide two explicit examples of the global residue condition that will be relevant to later sections. Example 2.1. Consider a genus g = 2 curve C with a P1-tail X, attached at a node x, a general point in C. Let j ≥ 2. Figure 1 depicts a twisted canonical divisor of type κ = (j,−j,1,1) on this 4 nodal curve with D = x+p ∼K , C 4 C D = jp −jp +p −3x∼K . X 1 2 3 X To find the conditions on such a twisted canonical divisor being smoothable we consider the X ≅P1 p 1 p 2 p 3 p 4 C x Figure 1: A twisted canonical divisor of type κ=(j,−j,1,1) on a nodal curve. conditions placed on the residues by the possible level graphs. In this case there are only two components and there is just one possible level graph pictured in Figure 2. The global residue condition then gives that the condition on the twisted differential being the limit of differentials on smooth curves is that there exists a differential ω with (ω )∼D and res (ω )=0. X X X x X C X Figure 2: The level graph giving the global residue condition. Consider the differential ω . By the cross ratio we can set the poles to 0 and ∞ and the zeros X to 1 and a. The resulting differential is given locally at 0 by (z−1)j(z−a) c dz z3 for some constant c∈C∗. The residue at 0 is j c(−1)j−1(j+( )a), 2 which is zero for the unique value a = −2/(j −1). Hence though any value of a gives a twisted canonical divisor, it is only for this unique value of a that the twisted canonical divisor is the limit of canonical divisors of this type on smooth curves. The necessity of the condition on the residue in this case can be observed by considering topo- logically a family of twisted canonical divisors on smooth curves degenerating to a nodal marked curve. Let χ be a family of meromorphic differentials (Z ,ω ) with ω of type κ = (j,−j,1,1) on t t t 5 p 1 p X 3 t p 2 v C p t 4 Figure 3: A twisted canonical divisor of type κ=(j,−j,1,1) on a smooth curve. smooth curves Z for t ≠ 0, degenerating to the nodal curve Z = C ∪ X at t = 0. Figure 3 shows t 0 x (Z ,ω ) for t ≠ 0. Let v be the vanishing cycle on Z that shrinks to the node x at t = 0. Observe t t t v separates the curve into components C and X with C → C and X → X as t → 0. Now as C t t t t t contains no poles, by an application of Stokes formula to the cycle v we have ∫ ωt =0. v The global residue condition is the limit of this condition as t → 0 and hence we have shown that the condition is necessary. Complex-analytic plumbing techniques and flat geometry are used in [BCGGM] to show the global residue condition obtained in this way is sufficient in all cases. Example 2.2. The following example provides further practice working with the global residue condition and will also be relevant to our later calculations. Consider a genus g = 3 curve C with p 8 p 1 p 7 p Y ≅P1 2 p x X ≅P1 3 y p 4 p 5 p 6 C y+p +p +p ∼K 4 5 6 C Figure 4: A twisted canonical divisor of type κ=(d1,d2,d3,14,−1) on a nodal genus g=3 curve. 6 P1-tails we denote X and Y attached at a nodes x and y respectively. Figure 4 depicts a twisted canonical divisor of type κ = (d ,d ,d ,14,−1) with ∑d = 1 for d ≠ 0, {d } ≠ {1,1,−1} on this 1 2 3 i i i nodal curve with D = y+p +p +p ∼K C 4 5 6 C D = p −p −2x∼K X 7 8 X D = d p +d p +d p −3y ∼K . Y 1 1 2 2 3 3 Y To find the conditions on such a twisted canonical divisor being smoothable we consider the con- ditions placed on the residues by the possible level graphs. Observe that due to the simple pole Graph A Graph B C C Y X Y X Figure 5: Level graphs giving the global residue condition. at p , for any differential ω on X with (ω ) ∼ p −p −2x we have res (ω ) = −res (ω ) ≠ 0. 8 X X 7 8 x X p8 X Hence Figure 5 gives the two possible level graphs to provide smoothable twisted canonical divisors p p 8 2 p p p 7 1 3 Y X t t v v 2 1 p p 4 6 C t p 5 Figure 6: Canonical divisor of type κ=(d1,d2,d3,14,−1) on a smooth curve. 7 of this type. Graph A gives the condition res (ω )+res (ω )=0, x X y Y while Graph B gives the condition res (ω )=0. y Y Hence all configurations of y,p ,p ,p on the rational tail Y are smoothable. 1 2 3 The necessity of the condition on the residue in this case can be again observed by considering topologicallyafamilyoftwistedcanonicaldivisorsonsmoothcurvesdegeneratingtoanodalmarked curve. This situation is depicted in Figure 6. The different possible global residue conditions distinguish the relative speed at which v and v degenerate to nodes. Graph A in Figure 5 1 2 represents the situation where v and v degenerate to nodes at the same speed, while Graph B in 1 2 Figure 5 represents the situation that v degenerates faster than v , and hence due to the poles on 1 2 Y , this results in no residue condition at x. t 2.3. Divisor theory on M . Let λ denote the first Chern class of the Hodge bundle on M g,n g,n and ψ denote the first Chern class of the cotangent bundle on M associated to the ith marked i g,n point where 1 ≤ i ≤ n. These classes are extensions of classes defined on M that generate g,n Pic(M )⊗Q. The boundary ∆ = M −M of M is codimension one. Define ∆ as the g,n g,n g,n g,n 0 locus of curves in M with a non-separating node. Define ∆ for 0≤i≤g, S ⊆{1,...,n} as the g,n i∶S locus of curves with a separating node, separates the curve into a genus i component containing the marked points from S and a genus g −i component containing the marked points from Sc, the complement of S. Hence we require ∣S∣ ≥ 2 for i = 0 and ∣S∣ ≤ n−2 for i = g and observe that ∆ = ∆ . These boundary divisors are irreducible and can intersect each other and i∶S g−i∶Sc self-intersect. The class of ∆ in Pic(M )⊗Q is denoted by δ . i∶S g,n i∶S These divisor classes freely generate Pic(M )⊗Q for g ≥ 3. For g = 2 the classes λ,δ and δ g,n 0 1 generate Pic(M )⊗Q with the relation 2 1 1 λ= δ + δ . 0 1 10 5 Pulling back this relation under the map ϕ ∶ M —→ M that forgets the n marked points we 2,n 2 obtain 1 1 λ= δ + ∑δ . 0 1∶S 10 5 1∈S Pic(M )⊗Qisfreelygeneratedbyλ,ψ andδ withthisrelation.[AC][HMo]giveanintroduction 2,n i i∶S to the divisor theory of M . g,n 3. Divisors from the strata of abelian differentials Definition 3.1. The divisor Dn in M for κ=(k ,...,k ) with ∑k =2g−2 is defined as κ g,n 1 m i Dn ={[C,p ,...,p ]∈M ∣ [C,p ,...,p ]∈M with ∑k p ∼K }, κ 1 n g,n 1 m g,m i i C where m=n+g−2 or n+g−1 for holomorphic and meromorphic signature κ respectively. When all k are even this divisor has two irreducible components based on spin structure that we denote i by the indices odd and even. Hence Dn is proportional to the pushdown of P(κ) under the map κ forgetting the last g−2 or g−1 points in the holomorphic or meromorphic cases respectively. We use this notation to denote both the divisor and the class. For holomorphic signature κ = (d ,...,d ,1g−2) with d > 0, the divisors Dn correspond to gen- 1 n i κ eralised Brill-Noether divisors which [L] used to investigate the Kodaira dimension of M . The g,n divisor Dn in M for the signature κ = (12g−2) is extremal as it is contracted by the map to κ g,g the universal Picard variety of degree g line bundles [FV1]. The closure of the anti-ramification 8 locus or the divisor Dn in M for the signature κ=(12g−4,2) was shown to be extremal by the κ g,g−1 construction of a covering curve with negative intersection [FV2]. The divisors of interest to us in this paper are those from meromorphic signature κ. In this section we summarise the known classes of these divisors. When all unmarked points are simple zeros the class was computed by [Mu¨][GZ]1. For d = (d ,...,d ) with ∑d = g−1 and at least one 1 n i d <0, i n d +1 ∣d −i∣+1 d −i+1 Dn =−λ+∑( j )ψ −0⋅δ − ∑ ( S )δ − ∑ ( S )δ d,1g−1 2 j 0 2 i∶S 2 i∶S j=1 i,S i,S S⊂S+ S⊄S+ in Pic(M )⊗Q, where S ∶={j ∣d >0} and d ∶=∑ d . g,n + j S j∈S j In [M2] the author generalises these to obtain for any fixed g ≥ 2 and n ≥ 2, a number of new infinite families of effective divisors from the strata of abelian differentials. The divisors relevant to our discussion are from meromorphic signatures with n ≥ 3. When the signature of unmarked points is (2g−1) we obtain the coupled partition divisors. For d=(d ,...,d ) such that ∑d =0 with 1 n i d− ≠{−2}, then n g g−1 Dn =2g−2(2g+1λ+2g−1∑d2ψ −2g−2δ − ∑ ∑2g−i+1(2i−1)δ −2g−1 ∑ ∑d2δ ). d,2g−1 j j 0 i∶S S i∶S j=1 ∣dS∣=0 i=0 ∣dS∣≥1i=0 ∣S∣≠n If all d are even then j 2g−1 n g 2g−1 g−1 Dn,odd =2g−2((2g−1)λ+ ∑d2ψ −2g−3δ − ∑ ∑(2i−1)(2g−i+1)δ − ∑ ∑d2δ ) d,2g−1 4 j j 0 i∶S 4 S i∶S j=1 ∣dS∣=0 i=0 ∣dS∣≥2i=0 ∣S∣≠n and 2g+1 n g 2g+1 g−1 Dn,even =2g−2((2g+1)λ+ ∑d2ψ −2g−3δ − ∑ ∑(2i−1)(2g−i−1)δ − ∑ ∑d2δ ). d,2g−1 4 j j 0 i∶S 4 S i∶S j=1 ∣dS∣=0i=0 ∣dS∣≥2i=0 For d=(−2,1,1), g−1 D3 = 2g−3(2g+1λ+2g+2ψ +2g−1(ψ +ψ )−2g−2δ − ∑2i+1(2g−i−1)δ d,2g−1 1 2 3 0 i{1,2,3} i=0 g−1 g−1 −∑2i+1(2g−i+1)δ − ∑2g−1(δ +δ ). i∶{2,3} i∶{1,2} i∶{1,3} i=0 i=0 When the signature of the unmarked points is (1g−2,2) we obtain the pinch partition divisors. For d=(d ,...,d ) with ∑d =g−2, d ≤−2 and d ≥0 for i≠j, then for d ≤−3, 1 n i j i j n g−1 Dn = (26−4g)λ+∑2d ((g−1)d +g−2)ψ −2δ + ∑c δ d,1g−2,2 i i i 0 i∶S i∶S i=1 i=0 where for j ∉S and d ≤i−1, S c =(2−2g)d2 +2(2gi+g−4i+1)d −2(gi2+gi−3i2+i+1) i∶S S S and for d ≥i, S c =(2−2g)d2 +2(2gi−g−4i+2)d −2(gi2−3i2−gi+4i). i∶S S S 1Note that in this formula S≠{1,...,n}. In this case the coefficient is found by Sc=∅⊂S+. The condition on S in the formula is separating the cases where all poles lie on the same component. 9 For d =−2 j (4g(d +1)−5d −9)d g−1 Dn = (27−4g)λ+4gψ +∑ i i iψ −2δ + ∑c δ d,1g−2,2 j 2 i 0 i∶S i∶S i≠j i=0 where for j ∉S and d ≤i−1, S 1 c = ((5−4g)d2 +(8gi+4g−18i+3)d −4gi2−4gi+13i2−3i−4) i∶S 2 S S and for d ≥i, S 1 c = ((5−4g)d2 +(8gi−4g−18i+9)d −4gi2+4gi+13i2−17i). i∶S 2 S S In the case that g = 2 the pinch partition and coupled partition divisors correspond. If further, all d are even, the coupled partition divisor formula gives the classes of each of the irreducible i components based on spin structure. 4. Moving curves A moving curve B of a projective variety X is a curve with B⋅D≥0 for all effective divisors D. Hence any effective divisor D with zero intersection with a moving curve must lie on the boundary of the effective cone as B ⋅(D−(cid:15)A) < 0 for any ample divisor A with (cid:15) > 0. One way to obtain movingcurvesisbyfibrations. IfthenumericalequivalenceclassesofB coveraZariskidensesubset of X and the general curve is irreducible then B is a moving curve. Fix a meromorphic signature κ of length m ≥ g+1. In §2.1 we introduce the subvariety P(κ) of M of codimension g. If ϕ ∶ P(κ) —→ M forgets the first g +1 points, we obtain a g,m g,m−g−1 fibration of P(κ) with one dimensional fibres. Consider the curve Bn defined as κ Bn ∶={[C,p ,...,p ]∈M ∣ fixed general [C,p ,...,p ]∈M and [C,p ,...,p ]∈P(κ)}. κ 1 n g,n g+2 m g,m−g−1 1 m Ifπ ∶P(κ)—→M forgetsallbutthefirstnpoints,thenforn≥g+1weobtainπ ϕ∗[C,p ,...,p ]= g,n ∗ g+2 m Bn. When n≤g we have π ϕ∗[C,p ,...,p ]=dBn for some positive integer d. κ ∗ g+2 m κ Proposition 4.1. The class of Bn is a moving curve in M when ∣κ∣ ≥ n + g and P(κ) is κ g,n irreducible. Proof. For any general [C,p ,...,p ] ∈ M to be in a numerical equivalence class of a curve Bn 1 n g,n κ we require some p ,...,p such that n+1 m m ∑k p ∼K . i i C i=1 Let d=∑n d and consider the map i=1 i f ∶Cm−n —→ Picd(C) (p ,...,p ) z→ K (−∑m d p ). n+1 m C i=n+1 i i The domain has dimension m−n, while the target has dimension g. Clearly f−1(∑n d p ) will be i=1 i i non-empty for general [C,p ,...,p ] when m ≥ n+g. Hence in these cases curves with numerical 1 n equivalence class equal to Bn cover a Zariski dense subset of M . κ g,n If a general curve with numerical class Bn is irreducible then Bn is a moving curve. However, if κ κ the general Bn is reducible, it can also be the class of a moving curve if all components of Bn have κ κ the same class and are hence proportional to Bn. κ InourcaseifageneralcurvewithnumericalclassBn isreduciblewithcomponentswithdifferent κ class, taking the closure of these distinguishable components over all [C,p ,...,p ] ∈ M n+1 m g,m−n would contradict the irreducibility of P(κ). (cid:3) 10