On the effective cone of higher codimension cycles in M ¯ ¯ ¯ ¯ ¯ ¯ g,n
aa r X i v : . [ m a t h . AG ] O c t ON THE EFFECTIVE CONE OF HIGHER CODIMENSION CYCLES IN M g,n SCOTT MULLANE
Abstract.
We exhibit infinitely many extremal effective codimension- k cycles in M g,n in the cases ● g ≥ , n ≥ g − k = ● g ≥ k ≤ n − g, g, and ● g = k ≤ n − Introduction
The birational geometry of the moduli space of curves is broadly dictated by the effective coneof divisors, which has attracted much attention [HMu][EH][F][FPop][CC1][M2]. Though compar-atively little is known, there has recently been growing interest in understanding finer aspects ofthe birational geometry encoded in the cones of higher codimension cycles [CC2][FL1][FL2][CT].In this paper we use meromorphic differentials on curves to construct infinitely many cycles thatform extremal rays of the effective cone of higher codimension cycles of M g,n for fixed g and n .Hence in these cases we show that the effective cone is not rational polyhedral.The meromorphic strata of canonical divisors of type κ = ( k , . . . , k n ) , P( κ ) ∶= { [ C, p , . . . , p n ] ∈ M g,n ∣ ∑ k i p i ∼ K C } for κ a meromorphic partition of 2 g −
2, form codimension g subvarieties in M g,n . When g = n ≥
3, in infinitely many cases the closure produces rigid and extremal divisors in M ,n and hence the effective cone of divisors is not rational polyhedral in these cases. In the casethat g ≥
2, the author [M2] showed the closure of infinitely many of these higher codimension cyclespushforward under the morphism forgetting marked points to give rigid and extremal divisors in M g,n for n ≥ g +
1. Hence in these cases the effective cones of divisors are not rational polyhedral.Chen and Coskun [CC2] showed higher codimension boundary stratum to be extremal in somecases and in the cases M ,n for n ≥ M ,n for n ≥ M , from the Keel-Vermeire divisors in M , . From the interior of the moduli space, Chen and Coskun [CC2] identified the closure of thelocus of hyperelliptic curves with a marked Weierstrass point in M , and the closure of the locusof hyperelliptic curves in M as extremal codimension two cycles. Chen and Tarasca [CT] showedthat for 1 ≤ n ≤
6, marking n Weierstrass points on a curve gave an extremal codimension n cyclein M ,n . Blankers [Bl] extended this to include marking any combination of Weierstrass pointsand pairs of points that are conjugate under the hyperelliptic involution. However, for any fixed n this still only produced finitely many extremal higher codimension cycles coming from the interiorof M ,n . Date : July 23, 2018. n this paper we use three methods to construct infinitely many extremal higher codimensioncycles from the strata of canonical divisors. In § § § κ for when the strata of canonical divisors P ( κ ) and pushfowards of this cycle forgetting marked points give extremal higher codimension cycles.In § g = § M g,n for g ≥ n ≥ g − Theorem 1.1. Eff ( M g,n ) is not rational polyhedral for g ≥ and n ≥ g − . We follow the general strategy of Chen and Coskun [CC2]. Consider the gluing morphism α ∶ ̂ ∆ ∶∅ = M g − ,n + × M , Ð → M g,n for g ≥
3, which glues a [ C, p , . . . , p n + ] ∈ M g − ,n + to [ E, q ] ∈ M , by identifying p n + with q toform a node. Proposition 3.1 shows pulling back extremal divisors on M g − ,n + provides extremaldivisors on ̂ ∆ ∶∅ . Further, the image of α is the loci contracted by the morphism ps ∶ M g,n Ð→ M ps g,n that contracts unmarked elliptic tails to cusps, where M ps g,n is the alternate compactification of M g,n by pseudo-stable curves. In this situation Proposition 3.2 shows these cycles will pushforward toprovide extremal codimension two cycles in M g,n provided α ∗ ∶ A ( ̂ ∆ ∶∅ ) = N ( ̂ ∆ ∶∅ ) Ð→ N ( M g,n ) is injective. Proposition 3.4 shows this map to be injective in the cases considered by showing thereare no nontrivial relations between the images of the generators of N ( ̂ ∆ ∶∅ ) through the use oftest surfaces in M g,n and pushing forward any possible such relation under forgetful morphismsand ps that contracts unmarked elliptic tails.For the rest of the paper we turn to cycles intersecting the interior of the moduli space. In § Theorem 1.2.
The cycle [ ϕ j ∗ P ( g − )] for j = , . . . , g − is rigid and extremal where ϕ j ∶M g, g − Ð→ M g, g − j − forgets the last j points. This extends the result of Farkas and Verra [FV] on divisors, which is used as the base case inan inductive argument similar to that used by Chen and Tarasca [CT]. This method is developedto more complicated situations in later sections. Assume the cycle [ ϕ j + ∗ P ( g − )] is rigid andextremal and let [ ϕ j ∗ P ( g − )] = ∑ c i [ V i ] be an effective decomposition with c i > V i irreducible codimension one subvarieties distinctfrom ϕ j ∗ P ( g − ) . Pushing forward under the map π m ∶ M g, g − j − Ð→ M g, g − j − forgetting the m th marked point for m = , . . . , g − j − [ ϕ j + ∗ P ( g − )] andhence there is some l with π m ∗ [ V l ] = k [ ϕ j + ∗ P ( g − )] for 0 < k ≤
1. But as ϕ j + ∗ P ( g − ) is rigid, this implies that V l is supported in π − m ( ϕ j + ∗ P ( g − )) .Further, any such cycle must push forward under the map forgetting any of the marked points to givea non-zero cycle, which must then be proportional to the rigid and extremal cycle [ ϕ j + ∗ P ( g − )] and we obtain that V l is supported on g − − j ⋂ m = π − m ( ϕ j + ∗ P ( g − )) = ϕ j ∗ P ( g − ) roviding a contradiction and proving the theorem.In § Theorem 1.3.
For g ≥ the cycle [ ϕ j ∗ P ( d , d , d , g − )] for j = , . . . , g − is extremal and rigid,where ϕ j ∶ M g, g Ð→ M g, g − j forgets the last j points with d + d + d = , ∑ d i < d i ≤ − and some d i = if g = . This gives the following corollary on the structure of the effective cone.
Corollary 1.4. Eff k ( M g,n ) is not rational polyhedral for g ≥ and k ≤ n − g, g . The rigid and extremal divisors of [M2] are used as a base case in the inductive proof thatemploys the inductive strategy of the proof of Theorem 1.2. One complication occurs in the caseof g = V l discussed above. This complication is overcome by observing that the cycle [ ϕ j ∗ P ( g − ) − c l V l ] is effective. Hence pushing this cycle forward under forgetful morphisms in the cases of interestresults in an effective divisor which must have non-negative intersection with the covering curvesintroduced in § § g = m ≥ d j = ( d j , . . . , d jn − m + ) for j = , . . . , m be distinct non-zero integer partitions of zero. We define X ( d , . . . , d m ) ∶= {[ E, p , . . . , p n ] ∈ M ,n ∣ [ E, p , . . . , p n − m , p n − m + j ] ∈ P ( d j )} . Under certain conditions, we can show irreducibility.
Proposition 1.5. X ( d , . . . , d m ) is irreducible if gcd ( d j ) = for j = , . . . , m and d jn − m + = for j = , . . . , m . By the inductive strategy of earlier sections we obtain.
Theorem 1.6.
Let m ≥ and n ≥ m + , then X ( d , . . . , d m ) is rigid and extremal if gcd ( d j ) = for j = , . . . , m and d jn − m + = for j = , . . . , m . This immediately gives the following corollary on the structure of the effective cones.
Corollary 1.7.
The effective cone of codimension k cycles in M ,n is not rational polyhedral for k ≤ n − . Preliminaries
Subvarieties from the strata of canonical divisors.
The subvarieties of interest in thispaper are the stratum of canonical divisors with signature κ defined as P ( κ ) ∶= {[ C, p , ..., p n ] ∈ M g,n ∣ k p + ... + k n p n ∼ K C } . The codimension of P ( κ ) is g − g for κ holomorphic (all k i ≥
0) or meromorphic (some k i < ϕ S ∶ M g,n Ð→ M g,n −∣ S ∣ for S ⊆ { , ..., n } be themap that forgets the marked points indexed by S . For ease of notation we will let ϕ j denote themap that forgets the last j points, that is, ϕ j = ϕ S for S = { n − j + , ..., n } . Further, we will let π j denote the map that forgets only the j th point, that is, π j = ϕ S for S = { j } . espite many remaining interesting questions, the codimension one case is well studied. Weobtain a codimension one subvariety or divisor from P ( κ ) in the moduli space of marked genus g curves by forgetting marked points. The divisor D nκ in M g,n for κ = ( k , ..., k n + s ) with ∑ k i = g − D nκ = {[ C, p , ..., p n ] ∈ M g,n ∣ [ C, p , ..., p n + s ] ∈ M g,n + s with ∑ k i p i ∼ K C } , where s = g − g − κ respectively. Hence D nκ isproportional to [ ϕ s ∗ P ( κ )] .2.2. Degeneration of differentials.
A stable pointed curve [ C, p , ..., p n ] ∈ M g,n is contained in ̃ P ( κ ) , the moduli space of twisted canonical divisors of type κ = ( k , ..., k n ) as defined by Farkasand Pandharipande [FP] if there exists a collection of (possibly meromorphic) divisors D j ∼ K C j on each irreducible component C j of C such that(a) The support of D j is contained in the set of marked points and the nodes lying in C j ,moreover if p i ∈ C j then ord p i ( D j ) = k i .(b) If q is a node of C and q ∈ C i ∩ C j then ord q ( D i ) + ord q ( D j ) = − q is a node of C and q ∈ C i ∩ C j such that ord q ( D i ) = ord q ( D j ) = − q ′ ∈ C i ∩ C j we have ord q ′ ( D i ) = ord q ′ ( D j ) = −
1. We write C i ∼ C j .(d) If q is a node of C and q ∈ C i ∩ C j such that ord q ( D i ) > ord q ( D j ) then for any q ′ ∈ C i ∩ C j we have ord q ′ ( D i ) > ord q ′ ( D j ) . We write C i ≻ C j .(e) There does not exist a directed loop C ⪰ C ⪰ ... ⪰ C k ⪰ C unless all ⪰ are ∼ .Farkas and Pandharipande showed that in addition to the main component P ( κ ) containing P ( κ ) ,this space contained extra components completely contained in the boundary of the moduli space.Bainbridge, Chen, Gendron, Grushevsky and M¨oller [BCGGM] provided the condition that atwisted canonical divisor lies in the main component. Let Γ be the dual graph of C . A twistedcanonical divisor of type κ is the limit of twisted canonical divisors on smooth curves if thereexists a collection of meromorphic differentials ω i on C i with ( ω i ) = D i that satisfy the followingconditions(a) If q is a node of C and q ∈ C i ∩ C j such that ord q ( D i ) = ord q ( D j ) = − q ( ω i ) + res q ( ω j ) = ∼ and ≻ , such that for any level L and any connected component Y of Γ > L thatdoes not contain a prescribed pole we have ∑ level ( q ) = L,q ∈ C i ∈ Y res q ( ω i ) = global residue condition .2.3. Rigid and extremal cycle classes.
For a projective variety X , let N k ( X ) denote the R -vector space of codimension- k cycles modulo numerical equivalence. The cycles in N k ( X ) that canbe written as a positive sum of effective cycles form a convex cone inside N k ( X ) known as the effective cone of codimension- k cycles denoted Eff k ( X ) .An effective codimension- k cycle Y is extremal or spans an extremal ray in the effective cone ifthe class Y cannot be written as a sum m Y + m Y of effective Y i with m , m > Y, Y and Y are all proportional classes. An effective cycle Y is rigid if every cycle with class mY issupported on Y for every positive integer m .The codimension one case is special, a curve B contained in an effective divisor D is known asa covering curve for D if irreducible curves with numerical class equal to B cover a Zariski dense ubset of D . Negative intersection by a covering curve is a well-known criterion for an irreducibleeffective divisor to be extremal and rigid.2.4. Rigid and extremal divisors.
In this section we collect the known results on rigid andextremal divisor classes relevant to our later arguments.The closure of the locus of g points on general genus g curves that sit in a hyperplane sectionof the canonical embedding form a divisor in M g,g . The class of this divisor was first calculatedby Logan [L] who used it to investigate the Kodaira dimension of M g,n . From our perspective thisdivisor is D g g − = ( g − ) ! ϕ g − ∗ P ( g − ) . Kontsevich and Zorich [KZ] showed P ( g − ) to be irreducible and hence ϕ j ∗ P ( g − ) is irreduciblefor j = , ..., g −
2. In the divisorial case, or the case j = g −
2, Farkas and Verra [FV] further showedthat this divisor is rigid and extremal through the construction of a covering curve with negativeintersection.
Proposition 2.1. D g g − is an rigid and extremal divisor in M g,g for all g ≥ .Proof. [FV] (cid:3) On an elliptic curve E the structure sheaf and canonical bundle coincide. Through the use ofcovering curves with negative intersection, Chen and Coskun [CC1] showed that the condition thatpoints on an elliptic curve satisfy certain equations under the group law formed rigid and extremaldivisors in M ,n for n ≥
3. From our perspective we state these results in the following proposition.
Proposition 2.2.
The divisors D nd ,...,d n are rigid and extremal in M ,n for gcd ( d , . . . , d n ) = and n ≥ .Proof. [CC1] (cid:3) In [M2], the author also used covering curves with negative intersection to exhibit infinitely manyrigid and extremal divisors in M g,n for g ≥ n ≥ g + Proposition 2.3.
The divisors D g + d ,d ,d , g − are rigid and extremal in M g,g + for d + d + d = , ∑ d i < d i ≤ − , for all g ≥ .Proof. [M2] (cid:3) Similarly, these divisors are simply pushforwards of strata of canonical divisors with meromorphicsignatures D g + d ,d ,d , g − = ( g − ) ! ϕ g − ∗ P ( d , d , d , g − ) . Boissy [Bo] showed P ( d , d , d , g − ) to be irreducible and hence ϕ j ∗ P ( d , d , d , g − ) is irre-ducible for j = , ..., g − Enumerative geometry on a general curve.
In this section we present results on finitemaps that will be used in enumerative calculations in later sections.For a general genus g = C and non-zero integers d , d consider the map f d ,d ∶ C × C Ð→ Pic d + d ( C )( q , q ) z→ O C ( d q + d q ) . roposition 2.4. For d ≠ ± d the map f d ,d is finite with degree d d . Further, f d ,d has simpleramification along the diagonal ∆ and the locus I of points ( q , q ) that are conjugate under theunique hyperelliptic involution of C . The intersection ∆ ∩ I is the six Weierstrass points of C andthe ramification order at these points is .For d = d the map f d ,d is generically finite with degree d d . Further, f d ,d has simpleramification along ∆ and contracts I .For d = − d the map f d ,d is generically finite with degree d d . Further, f d ,d has simpleramification along I and contracts ∆ .Proof. This generalises [CT]. Fix d , d and let f = f d ,d . Take a general point e ∈ C and considerthe isomorphism H ∶ Pic d + d ( C ) Ð→ J ( C ) L z→ L ⊗ O C ( − ( d + d ) e ) . Now let F = H ○ f . Then we have deg F = deg f and F ( q , q ) = O C ( d ( q − e ) + d ( q − e )) . Let Θ be the fundamental class of the theta divisor in J ( C ) . By [ACGH] § = g ! = O C ( k ( x − e )) for varying x ∈ C has class k Θ in J ( C ) . Hencedeg F = deg F ∗ F ∗ ([ O C ]) = deg ( d d Θ ) = d d Now consider the branch and exceptional locus of F . This is the genus g = § F around the points ofinterest. If f dω, f dω is a basis for H ( C, K C ) , then locally analytically the map becomes ( q , q ) z→ ( ∫ q e d f dω + ∫ q e d f dω, ∫ q e d f dω + ∫ q e d f dω ) modulo H ( C, K C ) . The map on tangent spaces at any fixed point ( q , q ) ∈ C × C is the Jacobianof F at the point, which is DF ( q ,q ) = ( f ( q ) f ( q ) f ( q ) f ( q )) ( d d ) Ramification or contraction in the map F occurs when the map on tangent spaces is not injectivewhich takes place at the points where rk ( DF ( q ,q ) ) <
2. The ramification index at a point ( q , q ) will be equal to the vanishing order of the determinant of DF ( q ,q ) at the point.This can be written locally analytically by the basis dω and ω ( ω − α ) dω where 0 is conjugate to α . In local coordinates with ( q , q ) = ( s, t ) we have DF ( q ,q ) = ( s ( s − α ) t ( t − α )) ( d d ) and det ( DF ( q ,q ) ) = d d ( s − t )( s + t − α ) . The loci s − t = s + t − α are ∆ and I respectively and intersect at the Weierstrass point s = t = α /
2. When these irreducible loci are contracted is clear from examining their image inPic d + d ( C ) . (cid:3) .6. Moving curves.
In addition to defining subvarieties and divisors, the strata of canonicaldivisors also yield interesting curves in M g,n . Taking a fibration of P ( κ ) for a meromorphicsignature κ we obtain the curve B nκ ∶= {[ C, p , ..., p n ] ∈ M g,n ∣ fixed general [ C, p g + , ..., p m ] ∈ M g,m − g − and m ∑ i = k i p i ∼ K C } . For m = ∣ κ ∣ ≥ n + g these curves provide moving curves in M g,n , that is, curves that have non-negativeintersection with all effective divisors. Proposition 2.5.
For g ≥ , meromorphic signature κ with ∣ κ ∣ = g + n − and n ≥ g + , B nκ, , − isa moving curve in M g,n . Further B nκ, , − ⋅ D nκ = . Hence all non-negative sums of the irreducible components of the divisor D nκ lie on the boundary ofthe closure of the effective cone known as the pseudo-effective cone.Proof. [M2, Theorem 1.1] (cid:3) In § B nκ, , − with certain boundary divisors for specific n and κ . We present the required intersection numbers in the following propositions. Proposition 2.6.
For κ = ( − h, , h ) with h ≥ , B κ, , − ⋅ δ ∶{ , } = h Proof.
To find B κ, , − ⋅ δ ∶{ , } we need to enumerate the limits of differentials of this signature with p and p sitting together on a rational tail. Hence we require the points p and p such that − hp + p + hq + q − q ∼ K C with p ≠ p , p i ≠ q j and any limits that may occur with these points colliding that will satisfy theglobal residue condition.To enumerate such points we consider the map f h, − ∶ C × C Ð→ Pic h − ( C )( p , p ) z→ O C ( hp − p ) . introduced in § hq − q + q − K C ∈ Pic h − ( C ) will provideus with the solutions of interest. By Proposition 2.4 for h ≥ h ,simply ramified along the diagonal ∆ and the locus of pairs of points that are conjugate underthe hyperelliptic involution denoted I . For h = h = I .For a general choice of q i the fibre will contain no solutions where p and p coincide with eachother or any of the q i . Hence we have found all solutions and B κ, , − ⋅ δ ∶{ , } = h . (cid:3) Proposition 2.7.
For κ = ( d , d , , d ) with d i ∈ Z ∖ { } , d ≥ and d + d + d = , B κ, , − ⋅ δ ∶{ , } = d d Proof.
To find B κ, , − ⋅ δ ∶{ , } we need to enumerate the limits of differentials of this signature with p and p sitting together on a rational tail. Hence we require the points p , p , p such that forfixed general q i , d p + d p + p + q + d q + q − q ∼ K C ith p i ≠ p j for i ≠ j and p i ≠ q j and any limits that may occur with these points colliding that willsatisfy the global residue condition.Consider the map f ∶ C Ð→ Pic − d ( C )( p , p , p ) z→ O C ( d p + d p + p ) . The fibre of this map above K C ( − q − d q − q + q ) ∈ Pic − d ( C ) will give us the solutions ofinterest. Take a general point e ∈ C and consider the isomorphism h ∶ Pic − d ( C ) Ð→ J ( C ) L z→ L ⊗ O C ( − de ) . Now let F = h ○ f , then deg F = deg f . Observe F ( p , p , p ) = O C ( d ( p − e ) + d ( p − e ) + ( p − e )) . Let Θ be the fundamental class of the theta divisor in J ( C ) . By [ACGH] § g = g ! = O C ( k ( x − e )) for varying x ∈ C has class k Θ in J ( C ) . Hencedeg F = deg F ∗ F ∗ ([ O C ]) = deg ( d Θ ⋅ d Θ ⋅ Θ ) = d d . As we have chosen the q i general, the general fibre will contain no points where the p i coincide witheach other or with the q i . Hence we have found all solutions and B κ, , − ⋅ δ ∶{ , } = d d . (cid:3) Extremal cycles supported in the boundary
In this section we investigate higher codimension effective cycles supported in the boundary of M g,n to show Eff ( M g,n ) is not finite polyhedral for g ≥ n ≥ g −
1. We follow the methodspresented in [CC2] using the infinitely many extremal effective divisors presented in [M2].Consider the gluing morphism α ∶ ̂ ∆ ∶∅ = M g − ,n + × M , Ð→ ∆ ∶∅ ⊂ M g,n for g ≥
3, which glues a [ C, p , . . . , p n + ] ∈ M g − ,n + to [ E, q ] ∈ M , by identifying p n + with q toform a node.Define the cycle Γ as M g − ,n + × ∆ and Γ i ∶ S , Γ K i , Γ λ , Γ as the pullback of δ i ∶ S , K i , λ and δ respectively, under the forgetful morphism ̂ ∆ ∶∅ = M g − ,n + × M , Ð→ M g − ,n + . Since M , ≅ P we have A ( ̂ ∆ ∶∅ ) ≅ N ( ̂ ∆ ∶∅ ) and is generated by the classes described abovefor g ≥
4. When g = . Consider the following proposition presented in [CC2]. Proposition 3.1.
Let X and Y be projective varieties such that numerical equivalence and ra-tional equivalence are the same for codimension k cycles in X , Y and X × Y respectively, with R -coefficients. Suppose Z is an extremal effective cycle of codimension k in X . Then Z × Y is anextremal effective cycle of codimension k in X × Y .Proof. [CC2, Corollary 2.4] (cid:3) ence pulling back the infinitely many extremal divisors presented in Proposition 2.3 providesinfinitely many extremal effective divisors in ̂ ∆ ∶∅ for g ≥ n ≥ g −
1. To show these cyclespushforward to provide extremal codimension-two cycles in M g,n we will require more machinery.For a morphism f ∶ X Ð→ Y between two complete varieties we associate an index to any Z , asubvariety of X , e f ( Z ) = dim ( Z ) − dim ( f ( Z )) . Proposition 3.2.
Let α ∶ Y Ð→ X be a morphism between two projective varieties. Assume that A k ( Y ) Ð→ N k ( Y ) is an isomorphism and that the composite α ∗ ∶ A k ( Y ) Ð→ A k ( X ) Ð→ N k ( X ) is injective. Moreover, assume that f ∶ X Ð→ W is a morphism to a projective variety W whoseexceptional locus is contained in α ( Y ) . If a k -dimensional subvariety Z ⊂ Y is an extremal cyclein Eff k ( Y ) and if e f ( α ( Z )) > , then α ( Z ) is also extremal in Eff k ( X ) .Proof. [CC2, Proposition 2.5] (cid:3) We apply this proposition to the situation Y = ̂ ∆ ∶∅ and X = M g,n where f is the morphism ps ∶ M g,n Ð→ M psg,n that contracts unmarked elliptic tails to cusps. Indeed, the exceptional locus of ps is ∆ ∶∅ . Itremains to show that α ∗ ∶ N ( ̂ ∆ ∶∅ ) Ð→ N ( M g,n ) is injective. To this end, we introduce a numberof test surfaces in M g,n .Consider the following test surfaces ● S a : Fix a general smooth curve [ C, p , . . . , p n + ] ∈ M g − ,n . Form the surface by attaching ageneral pencil of plane cubics at a base point to p n + to form a node and allowing the point p to vary in the pencil. ● S b ∶ P : Fix 1 ≤ i ≤ n and P ⊂ { , . . . , n + } with i, n + ∈ P . Fix general smooth curves [ X, q , . . . , q ∣ P ∣ ] ∈ M , ∣ P ∣ , [ C, q ′ , . . . , q ′ n −∣ P ∣+ ] ∈ M g − ,n −∣ P ∣+ and [ E, q ] ∈ M , . Form thesurface by attaching q ∣ P ∣ to q and q ∣ P ∣− to a point q ′ that varies freely in C , to form nodes.Label q , . . . , q ∣ P ∣− as p m for m ∈ P ∖ { i, n + } and q ′ , . . . , q ′ n −∣ P ∣+ as p m for m ∈ P c ∖ . Allowthe point p i to vary in E . ● S c : Fix distinct 1 ≤ i, j ≤ n . Fix general smooth curves [ C, q , . . . , q n − ] ∈ M g − ,n − and [ E, q ] ∈ M , . Form the surface by attaching q n − to q to form a node and labelling q , . . . , q n − as p m for m ∈ { , . . . , n } ∖ { i, j } . Allow p i and p j to vary freely in E and C respectively. ● S d ∶ P : Fix 1 ≤ i ≤ n and P ⊂ { , . . . , n + } such that i, n + ∈ P and n ≥ ∣ P ∣ ≥
4. Fixgeneral smooth curves [ X, q , . . . , q ∣ P ∣ ] ∈ M , ∣ P ∖{ i }∣ , [ C, q ′ , . . . , q ′ n −∣ P ∪{ i }∣ ] ∈ M g − ,n −∣ P ∪{ i }∣ and [ E, q ] ∈ M , . Form the surface by attaching the point q ′ n −∣ P ∪{ i }∣ to a point x that variesin X to form a node. Label the points q , . . . , q ∣ P ∖{ i }∣ as p m for m ∈ P ∖ { i } and the poins q ′ , . . . , q ′ n −∣ P ∪{ i }∣ as p m for m ∈ P c ∖ { i } . Attach the point q to p n + to form a node and allow p i to vary freely in E . ● S eh ∶ P : Fix 0 ≤ h ≤ g − ≤ i ≤ n and P ⊂ { , . . . , n + } such that j ∈ P and n + ∉ P and if h = ∣ P ∣ ≥
3, if h = ∣ P ∣ ≥ h = g − ∣ P ∣ ≤ n −
1. Fix generalsmooth curves [ C, q , . . . , q n −∣ P ∣+ ] ∈ M g − h − ,n −∣ P ∣+ and [ C ′ , q ′ , . . . , q ′∣ P ∣ ] ∈ M h, ∣ P ∣ . Form thesurface by identifying the point q n −∣ P ∣+ with the point q ′∣ P ∣ , and the point q n −∣ P ∣+ with apoint p that varies in C to form nodes. Label q , . . . , q n −∣ P ∣ as p m for m ∈ P c ∖ { n + } andlabel q ′ , . . . , q ′∣ P ∣− as p m for m ∈ P ∖ { i } . Let p i vary in C ′ . ● S fh : Fix distinct 1 ≤ i, j ≤ n and 0 ≤ h ≤ g −
2. Fix general smooth curves [ C , q , q ] ∈ M h, , [ C , q ′ , . . . , q ′ n ] ∈ M g − h − ,n and [ E, q ] ∈ M , . Form the surface by attaching q and q ′ n todistinct base points of a general pencil of plane cubics and q ′ n − to q to form nodes and abelling q as p j and q ′ , . . . , q ′ n − as p m for m ∈ { , . . . , n } ∖ { i, j } . Allow p i to vary freely in E . ● S g : Fix 1 ≤ i ≤ n and fix general smooth curves [ C, p, p , . . . , ˆ p i , . . . , p n + ] ∈ M g − ,n + and [ E, q ] ∈ M , . Form the surface by attaching a general pencil of plane quartics at a basepoint to p and attaching p n + to q to form nodes. Allow the point p i to vary freely in E . ● S h : Fix g ≥ ≤ i, j ≤ n and general smooth curves [ E ′ , p, q ] ∈ M , , [ C ′ , q ′ , . . . , q ′ n − ] ∈M g − ,n − and [ E, q ] ∈ M , . Form the surface by attaching p, q ′ n − and q to distinct basepoints of a general pencil of plane quartics to form nodes and labelling q = p j and q ′ , . . . , q ′ n − as p m for m ≠ i, j . Allow the point p i to move freely in E . Proposition 3.3.
Let γ, γ λ , γ , γ i ∶ S , γ K j denote the classes in M g,n of the cycles Γ , Γ λ , Γ , Γ i ∶ S , Γ K j respectively. The surfaces have the following intersection numbers. S a ⋅ γ = , S a ⋅ γ K n + = S b ∶ P ⋅ γ ∶ P ∪{ j } = for j ∈ P c , S b ∶ P ⋅ γ ∶ P = − ( g − ) − ( n + − ∣ P ∣) S b ∶ P ⋅ γ K j = ( g − ) − for j ∈ PS c ⋅ γ ∶{ j,m } = for m ≠ i, j, n + , S c ⋅ γ ∶{ i,n + } = − , S c ⋅ γ ∶{ i,j,n + } = , S c ⋅ γ K j = ( g − ) − ,S d ∶ P ⋅ γ ∶ P ∖{ j } = for j ∈ P , S d ∶ P ⋅ γ ∶ P = − ∣ P ∣ , S d ∶ P ⋅ γ ∶{ i,n + } = − , S d ∶ P ⋅ γ ∶ P ∖{ i,n + } = ,S d ∶ P ⋅ γ K j = − for j ∈ P ∖ { i, n + } , S d ∶ P ⋅ γ K j = − ∣ P ∣ for j = i, n + S eh ∶ P ⋅ γ h ∶ P = − , S eh ∶ P ⋅ γ h ∶ P ∖{ j } = , S eh ∶ P ⋅ γ ∶{ j,m } = for m ∈ P ∖ { j } S eh ∶ P ⋅ γ K j = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ( g − ) − for h = g − h − for h = , . . . , g − for h = S fh ⋅ γ h ∶{ j } = − , S fh ⋅ γ h + ∶{ j } = − , S fh ⋅ γ λ = , S fh ⋅ γ = ,S fh ⋅ γ K m = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ for h = , m = j for h = g − all m otherwise S g ⋅ γ λ = , S g ⋅ γ = , S g ⋅ γ ∶∅ , S g ⋅ γ K j = ⎧⎪⎪⎨⎪⎪⎩ for g = for all j for g ≥ S h ⋅ γ ∶{ j } = − , S h ⋅ γ g − ∶{ ,...,n }∖{ i,j } = − , S h ⋅ γ ∶{ i,n + } = − S h ⋅ γ K i = , S h ⋅ γ K n + = , S h ⋅ γ λ = , S h ⋅ γ = All other intersections are zero.Proof.
The intersection of each surface with ∆ , ∅ is transverse providing a curve to intersect withΓ , Γ λ , Γ , Γ i ∶ S , Γ K j inside ∆ , ∅ . The intersection numbers are a simple exercise in intersection the-ory [HMo]. (cid:3) Proposition 3.4. α ∗ ∶ N ( ̂ ∆ ∶∅ ) Ð→ N ( M g,n ) is injective for g ≥ and n ≥ g − .Proof. Consider a non-trivial relation on the classes in M g,n (1) cγ + c λ γ λ + c γ + ∑ i,S c i ∶ S γ i ∶ S + n + ∑ j = c K j γ K j = , here for g = c = λ and the boundary classes ingenus g =
2. By intersecting this equation with the surfaces presented in Proposition 3.3 we obtainrelations on the coefficients. The action of S n on M g,n permuting the marked points ensures thatany such nontrivial relation (1) will give a nontrivial relation such that c K i = c K j for i, j ≠ n + c i ∶ S = c i ∶ T for ∣ S ∣ = ∣ T ∣ and ∣ S ∪ { n + }∣ = ∣ T ∪ { n + }∣ . Hence we assume these symmetries hold anddenote c K = c K i for i ≠ n + c i ∶∣ S ∣ = c i ∶ S for n + ∉ S .Consider ps ∶ M g,n Ð→ M psg,n that contracts unmarked elliptic tails. All cycles in (1) except γ are contracted by this morphism. Hence pushing down the relation we obtain c γ = . Test surface S a then immediately implies c K n + = . Test surface S b ∶ P for P = { , . . . , n + } implies(2) ( ( g − ) − ) nc K − ( ( g − ) − ) c g − ∶ = , and(3) ( ( g − ) − )( n − k ) c K − ( ( g − ) − + k ) c g − ∶ k + kc g − ∶ k − = k = , . . . , n −
1. Test surface S c gives(4) ( ( g − ) − ) c K + ( n − ) c ∶ − c g − ∶ n − + c g − ∶ n − = . Now consider π i ∶ M g,n Ð→ M g,n − that forgets the i th marked point. Pushing forward we obtain π i ∗ γ ∶{ i,j } = δ , ∅ for j = , . . . , ˆ i, . . . , n + , π i ∗ γ K i = ( ( g − ) − ) δ , ∅ , with all other cycles pushing forward to give zero. Hence pushing forward (1) we obtain(5) ( ( g − ) − ) c K + ( n − ) c ∶ + c g − ∶ n − = . For g ≥ n = c ∶ = c g − ∶ = c g − ∶ = c K = . For g ≥ n ≥ g − , S d ∶ P for P = { , , , n + } and i = − c K + c ∶ − c g − ∶ n − + c g − ∶ n − − c g − ∶ n − = . In this case, equations (2),(3),(4),(5) and (6) provide independent relations, hence c ∶ = c g − ∶ n − = c g − ∶ n − = c g − ∶ n − = c K = . Hence by (2) and (3), c g − ∶ s = s = , . . . n − S eh ∶ P gives the relation between the remaining coefficients(7) c h ∶ s = c h ∶ s − where if h = ≤ s ≤ n , if h = ≤ s ≤ n , if h = g − ≤ s ≤ n − h = , . . . , g − ≤ s ≤ n . Hence c ∶ s = s = , . . . , n .Test surface S fh gives the relation(8) − c h ∶ − c h + ∶ + c λ + c = or 0 ≤ h ≤ g − c ∶ = g = c term. Comparing this equationfor consecutive h gives c h ∶ = c h + ∶ for 0 ≤ h ≤ g −
3. Combined with equation (7) this gives c h ∶ s = h with 0 ≤ s ≤ n . For g even this extends by symmetry to all h and s except h = s = g odd we have(9) c i ∶ s = c j ∶ t = c λ + c for i, j odd 0 ≤ s, t ≤ n but s, t ≠ i, j =
1. In the case g = c term and i = j = S g gives the relation(10) 3 c λ + c = . For g = c term and hence c λ = c ∶ s = ≤ s ≤ n . Hencethe one remaining coefficient in (1) must be c ∶ = α ∗ is injective for g = g ≥ c λ + c = . Equation (10) then implies c λ = c = c ∶ = α ∗ is injective for even g ≥ g ≥ S h gives − c ∶ − c g − ∶ n − + c λ + c = . Hence (9) and (10) implies c i ∶ s = i, s except i = , s =
0. Further, (10) and (9) give c λ = c = c ∶ = α ∗ is injective for all g ≥ (cid:3) Theorem 1.1. Eff ( M g,n ) is not rational polyhedral for g ≥ and n ≥ g − .Proof. Consider the forgetful morphism ̂ ∆ ∶∅ = M g − ,n + × M , Ð→ M g − ,n + . Pulling back the infinitely many extremal divisors of Proposition 2.3 we obtain by Proposition 3.1infinitely many extremal divisors on ̂ ∆ ∶∅ for g ≥ n ≥ g −
1. Propositions 3.2 and 3.4 completethe proof. (cid:3) Principal strata
Theorem 1.2.
The cycle [ ϕ j ∗ P ( g − )] for j = , . . . , g − is rigid and extremal where ϕ j ∶M g, g − Ð→ M g, g − j − forgets the last j points.Proof. Proceed by induction. Assume that ϕ j + ∗ P ( g − ) is rigid and extremal. If [( ϕ j ) ∗ P ( g − )] is not extremal then it can be expressed as [( ϕ j ) ∗ P ( g − )] = ∑ c i [ V i ] for c i > V i irreducible with class not proportional to [( ϕ j ) ∗ P ( g − )] . Pushing forward thisequation by π k ∶ M g, g − j − Ð→ M g, g − j − forgetting the k th marked point for k = , . . . , g − j − ( π k ) ∗ [( ϕ j ) ∗ P ( g − )] = [( ϕ j + ) ∗ P ( g − )] = ∑ c i ( π k ) ∗ [ V i ] or each k = , ..., g − j − k there is at least one V i such that ( π k ) ∗ [ V i ] is non-zero.Further, as the LHS is extremal, ( π k ) ∗ [ V i ] is necessarily a positive multiple of [( ϕ j + ) ∗ P ( g − )] .As this cycle is rigid, V i must be supported on ( π k ) − ( ϕ j + ∗ P ( g − )) and hence ( π k ′ ) ∗ [ V i ] isnon-zero for any other k ′ . This argument for each k ′ yields V i is supported in the intersectionof ( π k ) − ( ϕ j + ∗ P ( g − )) for k = , ..., g − j −
2. In particular, any 2 g − j − V i are distinct points in a hyperplane section of the canonical embedding, hence all2 g − j − V i is supported on ( ϕ j ) ∗ P ( g − ) and hence is a positive multiple of [( ϕ j ) ∗ P ( g − )] providing acontradiction.Hence [ ϕ j ∗ P ( g − )] is extremal if [ ϕ j + ∗ P ( g − )] is rigid and extremal. Further, if [( ϕ j ) ∗ P ( g − )] is extremal but not rigid, then [( ϕ j ) ∗ P ( g − )] = c [ V ] for c > V not supported on ( ϕ j ) ∗ P ( g − ) . The above argument provides a contradiction.The base case of the inductive argument j = g −
2, is the divisorial case presented in Theorem 2.1 (cid:3) Meromorphic strata
The strategy employed to show the principal strata are rigid and extremal can be applied tothe meromorphic strata with an alteration for some lower genus cases. We provide the inductiveargument as the following series of propositions separating the more involved lower genus cases.
Proposition 5.1.
For g ≥ and j = , . . . , g − the cycle [ ϕ j ∗ P ( d , d , d , g − )] is extremal andrigid, where ϕ j ∶ M g, g Ð→ M g, g − j forgets the last j points with d + d + d = , ∑ d i < d i ≤ − andsome d i = if g = .Proof. Proceed again by induction. Assume [( ϕ j + ) ∗ P ( d , d , d , g − )] is rigid and extremal. If [( ϕ j ) ∗ P ( d , d , d , g − )] is not extremal then it can be expressed as [( ϕ j ) ∗ P ( d , d , d , g − )] = ∑ c i [ V i ] for c i > V i irreducible with class not proportional to [( ϕ j ) ∗ P ( d , d , d , g − )] . Pushing forwardthis equation under π k ∶ M g, g − j Ð→ M g, g − j − forgetting the k th point we obtain ( π k ) ∗ [( ϕ j ) ∗ P ( d , d , d , g − )] = [( ϕ j + ) ∗ P ( d , d , d , g − )] = ∑ c i ( π k ) ∗ [ V i ] for each k = , ..., g − j for g ≥
3. Without loss of generality assume that d = g =
3. Thenthe equation will hold in the g = k = , . . . , − j .As the LHS is non-zero, for a fixed k there is at least one V i such that ( π k ) ∗ [ V i ] is non-zero. Fur-ther, as the LHS is extremal, ( π k ) ∗ [ V i ] is necessarily a positive multiple of [( ϕ j + ) ∗ P ( d , d , d , g − )] .But as this cycle is rigid, V i must be supported on ( π k ) − ( ϕ j + ∗ P ( d , d , d , g − )) and hence ( π k ′ ) ∗ [ V i ] is non-zero for any other k ′ = , ..., g − j for g ≥ k ′ = , . . . − j for g =
3. Thisargument for each k ′ yields V i is supported in the intersection of ( π k ) − ( ϕ j + ∗ P ( d , d , d , g − )) for k = , ..., g − j for g ≥ k = , . . . − j for g =
3. A general element of V i is hence of the form [ C, p , ..., p g − j ] ∈ M g, g − j with d p + d p + d p + g − j ∑ i = ,i ≠ k p i + j + ∑ i = q i ∼ K C or some q i with k = , ..., g − j . But this implies that for g ≥ p i for i = , ..., g − j are allat least pairwise distinct and hence distinct. Similarly for g = p i for i = , . . . , − j are all atleast pairwise distinct and hence distinct.Hence in this case we have V i is supported on ( ϕ j ) ∗ P ( d , d , d , g − ) and [ V i ] is a positivemultiple of [( ϕ j ) ∗ P ( g − )] providing a contradiction.Hence if ( ϕ j + ) ∗ P ( d , d , d , g − ) is rigid and extremal then ( ϕ j ) ∗ P ( d , d , d , g − ) is rigidand extremal. The base case for the inductive argument is the divisorial case j = g − (cid:3) Proposition 5.2.
For g = and j = , , the cycle [ ϕ j ∗ P ( d , d , d , )] is extremal and rigid,where ϕ j ∶ M , Ð→ M , − j forgets the last j points with d + d + d = , ∑ d i < d i ≤ − .Proof. The case where some d i = d i ≠ d ≥
2. Again, we proceed by induction. Assume [( ϕ j + ) ∗ P ( d , d , d , )] isrigid and extremal. If [( ϕ j ) ∗ P ( d , d , d , )] is not extremal then it can be expressed as [( ϕ j ) ∗ P ( d , d , d , )] = ∑ c i [ V i ] for c i > V i irreducible with class not proportional to [( ϕ j ) ∗ P ( d , d , d , )] . Pushing forwardthis equation we obtain ( π k ) ∗ [( ϕ j ) ∗ P ( d , d , d , )] = [( ϕ j + ) ∗ P ( d , d , d , )] = ∑ c i ( π k ) ∗ [ V i ] for each k = , ..., − j . As the LHS is non-zero, this implies for a fixed k , there is at least one V i such that ( π k ) ∗ [ V i ] is non-zero. Further, as the LHS is extremal, ( π k ) ∗ [ V i ] is necessarily a positivemultiple of [( ϕ j + ) ∗ P ( d , d , d , )] . But as this cycle is rigid, V i must be supported on ( π k ) − ( ϕ j + ∗ P ( d , d , d , )) and hence ( π k ′ ) ∗ [ V i ] is non-zero for any other k ′ = , ..., − j . This argument for each k ′ yields V i is supported in the intersection of ( π k ) − ( ϕ j + ∗ P ( d , d , d , )) for k = , ..., − j .Hence for j = V i is of the form [ C, p , . . . , p ] ∈ M , with d p + d p + d p + ∑ i = ,i ≠ k p i + q k ∼ K C for some q k for each k = , ,
6. But this implies that the p i for i = , , V i is supported on P ( d , d , d , ) providing a contradictionand showing if [( ϕ ) ∗ P ( d , d , d , )] is rigid and extremal then [ P ( d , d , d , )] is rigid andextremal.In the remaining case j = V i is supported in the intersection of ( π k ) − ( ϕ ∗ P ( d , d , d , )) for k = ,
5. In this case there are two possible candidates for where the irreducible cycle V i issupported. The cycle V i is supported on either ϕ ∗ P ( d , d , d , ) , or on the cycle X ∶= ϕ ∗ ( ϕ ∗ P ( d , d , d , )) ⋅ δ ∶{ , } which can also be described as X ∶= {[ C, p , p , p , q ] ∪ q = x [ P , x, p , p ] ∈ M , ∣ [ C, p , p , p , q ] ∈ D d ,d ,d , } where [ P , x, p , p ] is a rational curve marked at three distinct points and D d ,d ,d , = ϕ ∗ P ( d , d , d , ) . The irreducibility of X follows from the irreducibility of P ( d , d , d , ) . Hence if V i is supportedon X then [ V i ] is proportional to [ X ] and π ∗ [ V i ] = eδ ∶{ , } or some e > [( ϕ ) ∗ P ( d , d , d , g − )] − c i [ V i ] is effective, by pushing down under the morphismthat forgets the first marked point we obtain the effective class π ∗ ([( ϕ j ) ∗ P ( d , d , d , g − )] − c i [ V i ]) = D d ,d , ,d − c i eδ ∶{ , } . However, by Proposition 2.5 and Proposition 2.7 we observe B κ, , − ⋅ ( D d ,d , ,d − c i eδ ∶{ , } ) = − d d c i e < , for κ = ( d , d , , d ) , which contradicts the moving curve B κ, , − introduced in § V i is not supported on X and must be sup-ported on ϕ ∗ P ( d , d , d , ) . Hence [ ϕ ∗ P ( d , d , d , )] is rigid and extremal if [ ϕ ∗ P ( d , d , d , )] is rigid and extremal. The base case for the inductive argument is the divisorial case j = (cid:3) Proposition 5.3.
For g = , the cycle [ P ( h, − h, , )] for h ≥ is extremal and rigid.Proof. [ π k ∗ P ( h, − h, , )] is rigid and extremal for k = , [ P ( h, − h, , )] isnot extremal then it can be expressed as [ P ( h, − h, , )] = ∑ c i [ V i ] for c i > V i irreducible with class not proportional to [ P ( h, − h, , )] . Pushing forward thisequation we obtain ( π k ) ∗ [ P ( h, − h, , )] = [( π k ) ∗ P ( h, − h, , )] = ∑ c i ( π k ) ∗ [ V i ] for k = ,
4. However, this implies there is some V i such that π ∗ [ V i ] = [ π ∗ V i ] ≠
0. But as [ π ∗ P ( h, − h, , )] is extremal (Theorem 2.3), [ π ∗ V i ] must be a positive multiple of [ π ∗ P ( h, − h, , )] .Further, as [ π ∗ P ( h, − h, , )] is rigid, V i must be supported on π − ( π ∗ P ( h, − h, , )) .Hence π ∗ [ V i ] ≠ V i must be supported on π − ( π ∗ P ( h, − h, , )) .The intersection π − ( π ∗ P ( h, − h, , )) ∩ π − ( π ∗ P ( h, − h, , )) has two irreducible components. V i is either supported on P ( h, − h, , ) or X ∶= π ∗ ( π ∗ P ( h, − h, , )) ⋅ δ ∶{ , } which can also be described as X ∶= {[ C, p , p , q ] ∪ q = x [ P , x, p , p ] ∈ M , ∣ [ C, p , p , q ] ∈ D h, − h, } where [ P , x, p , p ] is a rational curve marked at three distinct points and D h, − h, = ϕ ∗ P ( h, − h, , ) = π ∗ P ( h, − h, , ) . The irreducibility of X follows from the irreducibility of P ( h, − h, , ) . Hence if V i is supported on X then [ V i ] is proportional to [ X ] and π ∗ [ V i ] = eδ ∶{ , } for some e > [ P ( h, − h, , )] − c i [ V i ] is effective, by pushing down under the morphism that forgetsthe first marked point we obtain the effective class π ∗ ([ P ( h, − h, , ))] − c i [ V i ]) = D − h, , ,h − c i eδ ∶{ , } . However, by Proposition 2.5 and Proposition 2.6 B κ, , − ⋅ ( D − h, , ,h − c i eδ ∶{ , } ) = − h e < , or κ = ( − h, , , h ) which contradicts the moving curve B κ, , − introduced in § V i is not supported on X and must besupported on P ( h, − h, , ) providing a contradiction with the given effective decomposition. Hence [ P ( h, − h, , )] is rigid and extremal. (cid:3) We record the previous three propositions as the following theorem.
Theorem 1.3.
For g ≥ the cycle [ ϕ j ∗ P ( d , d , d , g − )] for j = , . . . , g − is extremal and rigid,where ϕ j ∶ M g, g Ð→ M g, g − j forgets the last j points with d + d + d = , ∑ d i < d i ≤ − and some d i = if g = . This immediately gives the following corollary on the structure of the effective cones.
Corollary 1.4. Eff k ( M g,n ) is not rational polyhedral for g ≥ and k ≤ n − g, g .Proof. The rigid and extremal cycles presented in Theorem 1.3 have non-proportional classes asthe pushforwards ( ϕ g − j − ) ∗ [( ϕ j ) ∗ P ( d , d , d , g − )] = [( ϕ g − ) ∗ P ( d , d , d , g − )] = ( g − ) ! D g + d ,d ,d , g − have non-proportional classes as divisors by Theorem 2.3. Hence we have infinitely many extremalrays for k = n − g .To extend the result, fix k and pull back these classes under the forgetful morphism ϕ ∶ M g,n Ð→ M g,g + k . All previous arguments hold for the pullbacks. (cid:3) Extremal cycles in genus one
In this section we examine the genus one case. In this case the meromorphic strata of canonicaldivisors have codimension one and to produce higher codimension cycles we intersect the pullbacksof strata under forgetful morphisms.
Definition 6.1.
Set m ≥ d j = ( d j , . . . , d jn − m + ) for j = , . . . , m be distinct non-zero integerpartitions of zero. Then X ( d , . . . , d m ) ∶= {[ E, p , . . . , p n ] ∈ M ,n ∣ d jn − m + p n − m + j + n − m ∑ i = d ji p i ∼ O E for j = , . . . , m } or alternatively X ( d , . . . , d m ) ∶= {[ E, p , . . . , p n ] ∈ M ,n ∣ [ E, p , . . . , p n − m , p n − m + j ] ∈ P ( d j )} has codimension- m in M ,n with closure in M ,n denoted X ( d , . . . , d m ) .We now specialise to the subvarieties of interest to us. Proposition 1.5. X ( d , . . . , d m ) is irreducible if gcd ( d j ) = for j = , . . . , m and d jn − m + = for j = , . . . , m .Proof. By forgetting the last m − X ( d , . . . , d m ) in M ,n we obtain P ( d ) , which isirreducible. But for every [ E, p , . . . , p n − m + ] ∈ P ( d ) , by the group law there is a unique p n − m + j = − n − m ∑ i = d ji p i which will in general be distinct from p i for i = , . . . , p n − m + j − , hence [ E, p , . . . , p n − m , p n − m + j ] ∈ P ( d j ) . or j = , . . . , m .But P ( d ) is irreducible as gcd ( d ) =
1. Hence X ( d , . . . , d m ) is irreducible. (cid:3) Theorem 1.6.
Let m ≥ and n ≥ m + , then X ( d , . . . , d m ) is rigid and extremal if gcd ( d j ) = for j = , . . . , m and d jn − m + = for j = , . . . , m .Proof. If [ X ( d , . . . , d m )] is not extremal then it can be expressed as [ X ( d , . . . , d m )] = ∑ c i [ V i ] for c i > V i irreducible with class not proportional to [ X ( d , . . . , d m )] . Let ϑ j = ϕ { n − m + ,... ̂ n − m + j,...,n } ,that is, ϑ j ∶ M ,n Ð→ M ,n − m + forgets all but the marked points 1 , . . . , n − m, n − m + j for j = , . . . , m . Pushing forward underthis map we obtain ϑ j ∗ [ X ( d , . . . , d m )] = ( d n − m + ) [ P ( d j )] = ∑ c i ϑ j ∗ [ V i ] . For fixed j this implies there must be some i such that ϑ j ∗ [ V i ] ≠
0. But as [ P ( d j )] is an extremaldivisor ϑ j ∗ [ V i ] must be a positive multiple of [ P ( d j )] and further as [ P ( d j )] is rigid, ϑ j ∗ [ V i ] mustbe supported in P ( d j ) and hence V i is contained in ϑ − j P ( d j ) .But this implies that ϑ k ∗ [ V i ] ≠ k = , . . . , m and hence, by the above argument, V i issupported in the intersection of ϑ − j P ( d j ) for j = , . . . , m .This implies V i is supported in X ( d , . . . , d m ) providing a contradiction. (cid:3) This immediately gives the following corollary on the structure of the effective cones.
Corollary 1.7.
The effective cone of codimension k cycles in M ,n is not rational polyhedral for k ≤ n − .Proof. The rigid and extremal cycles presented in Theorem 1.6 have non-proportional classes as ϑ j ∗ [ X ( d , . . . , d k )] = ( d n − k + ) [ P ( d j )] = ( d n − k + ) D n − k + d j . have non-proportional classes for [ X ( d , . . . , d k )] with distinct d j by Theorem 2.2. Pulling backthese cycles under the forgetful morphism extends the result from k = n − k ≤ n − (cid:3) References [ACGH] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I.
Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] , . Springer-Verlag,New York, 1985.[BCGGM] M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, and M. M¨oller, Compactification of strata of abeliandifferentials, arXiv:1604.08834.[Bl] V. Blankers, Hyperelliptic classes are rigid and extremal in genus two, arXiv:1707.08676[Bo] C. Boissy, Connected components of the moduli space of meromorphic differentials, Comm. Math. Helv. (2015)no. 2, 255–286.[CC1] D. Chen and I. Coskun, Extremal effective divisors on M ,n . Math. Ann. (2014), no. 3–4, 891–908.[CC2] D. Chen and I. Coskun. Extremal higher codimension cycles on moduli spaces of curves.
Proc. London Math.Soc. (2015), 181–204.[CT] D. Chen and N. Tarasca, Extremality of loci of hyperelliptic curves with marked Weierstrass points,
AlgebraNumber Theory (2016), no. 9, 1935–1948.[EH] D. Eisenbud, and J. Harris, The Kodaira dimension of the moduli space of curves of genus ≥ Invent. Math. (1987), no. 2, 359–387.[F] G. Farkas, The geometry of the moduli space of curves of genus 23. Math. Ann. (2000), no. 1, 43–65.[FP] G. Farkas and R. Pandharipande, The moduli space of twisted canonical divisors, with an appendix by F. Janda,R. Pandharipande, A. Pixton, and D.Zvonkine,
J. Institute Math. Jussieu, to appear arXiv:1508.07940 . FPop] G. Farkas and M. Popa, Effective divisors on M g , curves on K Journalof Algebraic Geometry (2005), 241–267.[FV] G. Farkas and A. Verra, The classification of universal Jacobians over the moduli space of curves,
Comment.Math. Helv. (2013), no. 3, 587–611.[FL1] M. Fulger, and B. Lehmann, Morphisms and faces of pseudo-effective cones
Proc. Lon. Math. Soc. (2016),no. 4, 651–676[FL2] M. Fulger, and B. Lehmann, Positive cones of dual cycle classes,
Alg. Geom. (2017), no. 1, 1–28[HMo] J. Harris and I. Morrison, Moduli of curves, Graduate Texts in Mathematics , Springer-Verlag New York,1998.[HMu] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves,
Invent. Math. (1982),no.1, 23–88.[KZ] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with pre-scribed singularities, Invent. Math. (2003), no. 3, 631–678.[L] A. Logan, The Kodaira dimension of moduli spaces of curves with marked points,
Amer. J. Math. (2003),no. 1, 105–138[M1] S. Mullane, Divisorial strata of abelian differentials,
Int. Math. Res. Notices (2017), 1717–1748.[M2] S. Mullane, On the effective cone of M g,n , Adv. in Math. (2017), 500–519.[S] L. Schaffler, On the cone of effective 2-cycles on M , , Eur. J. Math. (2015), no. 4, 669–694. Scott Mullane, Department of Mathematics, University of Georgia Athens, GA 30602 USA
E-mail address : [email protected]@uga.edu