Tautological pairings on moduli spaces of curves
aa r X i v : . [ m a t h . AG ] F e b TAUTOLOGICAL PAIRINGS ON MODULI SPACES OFCURVES
RENZO CAVALIERI AND STEPHANIE YANG
Abstract.
We discuss analogs of Faber’s conjecture for two nestedsequences of partial compactifications of the moduli space of smoothcurves. We show that their tautological rings are one-dimensional intop degree but do not satisfy Poincar´e duality.
The structure of the tautological ring of the moduli space of stable curvesis predicted by the
Faber conjecture , which states that R ∗ ( M g,n ) is Goren-stein with socle in dimension 3 g − n = dim M g,n . We break this statementinto two parts: Socle:
The tautological ring vanishes in high degree and it is one-dimensional in top degree.(1) R k ( M g,n ) = ( , if k > g − n Q , if k = 3 g − n Poincar´e duality:
For 0 ≤ k ≤ g − n , the bilinear pairing R k ( M g,n ) × R g − n − k ( M g,n ) → R g − n ( M g,n )(2) is non-degenerate.In [FP2], Faber and Pandharipande speculate that the tautological ringsof M ct g,n and M rt g,n , two partial compactifications of the moduli space ofcurves, satisfy analogous properties. While the socle statements have sincebeen proven in all instances, Poincar´e duality remains open. We give evi-dence that the two properties are not necessarily immediately correlated.We define two chains of partial compactifications (Defs. 1.2 and 1.3) M ct g,n = M λ g g,n ⊆ M λ g − g,n ⊆ · · · ⊆ M λ g,n ⊆ M g,n (3) M rt g,n = M ch g − g,n ⊆ M ch g − g,n ⊆ · · · ⊆ M ch g,n ⊆ M g,n (4)and define their tautological rings by restriction. The main results of thispaper address the analog of Faber’s conjecture for these spaces. The soclestatement extends but Poincar´e duality fails. Proposition 1.
For i = 1 , , . . . , g , (1) R k ( M λ i g,n ) = 0 for k > g − n − i and R g − n − i ( M λ i g,n ) ∼ = Q , (2) R k ( M ch i − g,n ) = 0 for k > g − n − i and R g − n − i ( M ch i − g,n ) ∼ = Q . Proposition 2.
For g ≥ and ( g, n ) = (2 , , the pairing (5) R g ( M λ g,n ) × R g − n ( M λ g,n ) → R g − n ( M λ g,n ) is not perfect. Proposition 3.
For any fixed g and ≤ i ≤ g − , and for every n ≫ ,either the tautological restriction sequence (6) R i − ( M g,n \ M λ i g,n ) −→ R g ( M g,n ) −→ R g ( M λ i g,n ) −→ is not exact in the middle, or the pairing (7) R g ( M λ i g,n ) × R g − − i + n ( M λ i g,n ) → R g − − i + n ( M λ i g,n ) is not perfect. The Chow ring is known to be tautological in codimensions 0 and 1 ([M]).Therefore Proposition 3 immediately implies:
Corollary 4. If g ≥ and i = 2 , then the pairing (7) is not perfect forevery n ≫ . We also show that for g ≥
2, the dimension of the kernel of the map(8) R g ( M λ g,n ) → hom (cid:16) R g − n ( M λ g,n ) , R g − n ( M λ g,n ) (cid:17) . becomes arbitrarily large as either g or n grows (Corollary 5).In Lemma 1.4 we show that for the extremal case i = g , M λ g g,n = M ct g,n and M ch g − g,n = M rt g,n . Our results therefore answer extensions of speculationsabout these spaces that appeared in [FP2]. The socle statements of theFaber conjectures for M rt g,n , M ct g,n , and M g,n were proved by Graber-Vakil([GV2, § § § M g was shown by Looijenga and Faber ([L;Fa, Theorem 2]). The Poincar´e duality property is only known for g = 0 byKeel ([Ke]), and for M g for g ≤
23 by Faber ([Fa]).
Remark 1.
The tautological restriction sequence for M g is exact in degrees ≥ g −
1, and exactness is conjectured in all degrees for the M rt g,n and M ct g,n ([FP3]). Remark 2.
It is easy to see that Poincare’ duality fails in arbitrary degreesby taking nonzero elements of high degree in the ideal generated by thekernel of the map (8).
Remark 3.
Note that R ∗ ( M λ , ) is Gorenstein. The socle dimension is 2and it is straightforward to check that the intersection matrix for the twogenerators of R ( M λ , ) is non-degenerate. Since π ∗ ( M λ , ) = M λ , , Propo-sition 2 shows that if M is a moduli space which satisfies Faber’s conjecture,its universal family does not necessarily satisfy Faber’s conjecture. Remark 4.
The classes λ i and ch i − vanish respectively on M g,n \ M λ i g,n and M g,n \ M ch i − g,n . This motivates our notation. AUTOLOGICAL PAIRINGS ON MODULI SPACES OF CURVES 3
This paper is organized as follows: in § Background
The tautological ring R ∗ ( M g,n ) is a natural subring of the Chow ring A ∗ ( M g,n ) elegantly defined in [FP3]: as g and n vary, the tautological ringsform the smallest system of Q -subalgebras of A ∗ ( M g,n ) that are closed underthe natural forgetful morphisms(9) π i : M g,n +1 → M g,n , and the gluing morphisms ι irr : M g − ,n +2 → M g,n (10) ι g ,n : M g ,n +1 × M g ,n +1 → M g + g ,n + n . (11)The tautological ring contains boundary strata δ Γ (the closure of the locusof curves whose dual graph is Γ), cotangent line classes ψ i , Mumford-Morita κ classes, chern classes λ i of the Hodge bundle.Much is known about the intersection theory of such classes. An excel-lent, albeit unfinished and unpublished, reference is [Ko]. Other referencesinclude [Fu], [HKK + ], and [M].The following formula will be used in the proof of Proposition 2. Lemma 1.1.
For any value of n for which the integrals are defined: Z M ,n κ n − = 1(12) Z M ,n κ n = 124(13) Z M ,n κ i κ n − i = (cid:18) n − i + 1 (cid:19) − Proof.
Equations (12) and (13) follow immediately from the pullback for-mula for κ classes, ([AC, Eq. (1.10)]),(15) κ a = π ∗ n +1 ( κ a ) − ψ an +1 . RENZO CAVALIERI AND STEPHANIE YANG Z M ,n κ i κ n − i − = Z M ,n (cid:0) π ∗ ψ i +1 n +1 (cid:1) κ n − i − (16) = Z M ,n +1 ψ i +1 n +1 (cid:0) κ n − i − − ψ n − i − n +1 (cid:1) (17) = Z M ,n +1 ψ i +1 n +1 (cid:0) π ∗ ψ n − i − n +2 (cid:1) − Z M ,n +1 ψ n − n +1 (18) = Z M ,n +2 (cid:0) ψ i +1 n +1 − D n +1 ,n +2 (cid:1) ψ n − i − n +2 − Z M ,n +1 ψ n − n +1 (19) = Z M ,n +2 ψ i +1 n +1 ψ n − i − n +2 − − Z M ,n +1 ψ n − n +1 (20) = (cid:18) n − i + 1 (cid:19) − . (21) (cid:3) Definition 1.2. M λ i g,n ⊆ M g,n is the locus of curves whose dual graph hasgenus ≤ g − i . Equivalently, M λ i g,n is the locus of curves where the sum ofthe geometric genera of the components is at least i . Definition 1.3. M ch i − g,n ⊆ M g,n is the locus of curves with at least onecomponent of genus at least i . Lemma 1.4. M λ g g,n = M ct g,n and M ch g − g,n = M rt g,n .Proof. The dual graph of any curve in M λ g g,n is connected of genus 0 andthus is a tree. Any curve in M ch g − g,n has at least one (hence exactly one)component of genus g ; the other components must necessarily form trees ofrational curves. (cid:3) Lemma 1.5.
For i = 1 , , . . . , g , (1) The class λ i vanishes on M g,n \ M λ i g,n . (2) The class ch i − vanishes on M g,n \ M ch i − g,n Proof.
The boundary stratum δ Γ is the image of the gluing morphism(22) δ Γ = im (cid:16) ι Γ : Y M g j ,n j + d j → M g,n (cid:17) . The Hodge bundle splits when restricted to δ Γ as in [FP, Eqs. (17) and (18)]:(23) ι ∗ Γ ( E ) = ⊕ E g j ,n j + d j ⊕ O n . To see (a), use the Whitney formula,(24) ι ∗ Γ ( c ( E )) = c ( ι ∗ Γ E ) = Y (cid:0) λ ,j + · · · + λ g j ,j (cid:1) where λ i,j is the i -th chern class of the Hodge bundle of the j -th factor M g j ,n j + d j . If δ Γ ∈ M g,n \ M λ i g,n , then P g j < i and term of degree i inequation (24) vanishes. AUTOLOGICAL PAIRINGS ON MODULI SPACES OF CURVES 511 1 1
Figure 1.
These two dual graphs differ by a Feynman move.To see (b), use the additivity of the chern character, on δ Γ :(25) ι ∗ Γ (ch i − ( E )) = ch i − ( ι ∗ Γ E ) = X j ch i − ,j where ch i − ,j is the (2 i − j -th factor M g j ,n j + d j . Since δ Γ ∈ M g,n \ M ch i − g,n , all g j < i and thusch i − = 0. (cid:3) We conclude this section by recalling theorem ⋆ by Graber-Vakil ([GV2]),which is a key ingredient in the proof of Proposition 1. Theorem ⋆ . Any tautological class of degree k on M g,n vanishes on theopen set consisting of strata satisfying (26) < k − g + 1 . Socle
In this section we prove Proposition 1. The strategy of the proof is natu-ral: theorem ⋆ forces tautological classes of high degree to be supported onstrata with many rational components. On the other hand, curves in M λ i g,n and M ch i − g,n satisfy geometric conditions that limit the number of rationalcomponents. These constraints imply high degree vanishing and force tau-tological classes in the socle degree to be supported on exactly one boundarystratum up to rational equivalence. Definition 2.1.
A Feynman move replaces a portion of a graph of type (a)on a dual graph with one of type (b) or (c), as illustrated below:(a) ij kl (b) ik jl (c) il jk
The half edges i , j , k , and l may be glued to other half edges to form edges(See Figure 2). Lemma 2.2.
If the dual graphs of two boundary strata differ by Feynmanmoves, then they are rationally equivalent.Proof.
This is immediate by noting that M , ∼ = P , so its boundary pointsare rationally equivalent. This equivalence is preserved under gluing mor-phisms. (cid:3) Remark 5.
It is a standard combinatorial fact that two trivalent graphswith same number of vertices and edges differ by a finite number of Feynmanmoves ([W]).
RENZO CAVALIERI AND STEPHANIE YANG
The proof of Proposition 1 now follows from some careful bookkeeping.
Part 1.
Theorem ⋆ implies that any class in R g − n − i + k ( M λ i g,n ) ( k ≥ g + n − − i + k rational components. Let δ Γ be any one such boundary stratum. By thedefinition of M λ i g,n , the sum of the genera of the vertices of graph Γ is atmost g − i .Stability implies that the incidence of any rational vertex is at least three,and there are at most n marked half-edges incident to these vertices. Thusthere are at least(27) 3(2 g + n − − i + k ) − n = 6 g + 2 n − i + 3 k half edges which must be glued to other half-edges. Since g Γ ≤ g − i , at least(28) 6 g + 2 n − − i + 3 k − g + n − − i + k ) = i + k half-edges must be glued to vertices of positive genus. The total genus ofthe curve represented by Γ is g , so the only possibility is k = 0, i.e. allrational vertices trivalent, and the remaining i half edges glued to i verticesof genus 1 and incidence 1. By Remark 5, any two such graphs differ by afinite number of Feynman moves, and hence represent rationally equivalenttautological classes. It is immediate to check that such graphs actually livein degree 3 g − n − i . (cid:3) Part 2.
This is similar to the previous proof, except now we require thegraph Γ to have at least one vertex of genus ≥ i . Theorem ⋆ forces any classin R g + n − − i + k ( M ch i g,n ) ( k ≥
0) to be supported on boundary strata with atleast 2 g + n − − i + k rational components, and stability implies that thereis at least three times as many half edges, only n of which are not glued tosome other half edge. The total genus of the graph must be g , the onlyconsistent possibility is for k = 0, i.e. all rational vertices are trivalent andthere is exactly one vertex of genus i and incidence 1. Again, all such stratarepresent rationally equivalent classes because any two graphs are equivalentup to a finite number of Feynman moves.The degree of such strata is 3 g + n − − i , which means that one must havea class of degree i − i vertex. Since R i − ( M rt i, ) ∼ = Q (this is the socle statement of Faber’s conjecture for M rt i, ), there is only onesuch nonzero class up to scalar multiple. (cid:3) Remark 6.
It is important to note that λ i (resp. ch i − ) does not vanish onthe unique generator of R top ( M λ i g,n ) (resp. R top ( M ch i − g,n )), and hence it canbe used as an evaluation class: multiplication by λ i gives an isomorphismbetween the socle and R g − n ( M g,n ).3. Failure of Poincar´e duality for M λ g,n In this section we construct counterexamples to the Poincar´e duality partof Faber’s conjectures for M λ g,n = M ch g,n for g ≥ g, n ) = (2 , AUTOLOGICAL PAIRINGS ON MODULI SPACES OF CURVES 71 · · · · · · · · · n n − i ψ i − · · · · · · n n − Figure 2.
Generators in socle degree for M λ i g,n and M ch i g,n .Choose any triple ( a, b, c ) of integers satisfying a + b + c = g , where a isnon-negative and b and c are positive. Choose any subset S of the n points.Let Γ( a, b, c, S ) ∈ R g ( M g,n ) denote the graph with two vertices connectedby b edges: one genus 1 vertex with a self-edges and carrying the points in S , and one genus 0 vertex with c self-edges and carrying the points in S c (see Figure 3). Let δ ( a, b, c, S ) denote the associated boundary stratum.When ( a, c, S ) = ( c − , a + 1 , S c ), the strata δ ( a, b, c, S ) and δ ( c − , b, a +1 , S c ) are not rationally equivalent, but their difference lies in the kernel ofmultiplication by λ . Proof of Proposition 2.
Let γ = δ ( a, b, c, S ) and γ = δ ( c − , b, a + 1 , S c ).Note that if ( a, c, S ) = ( c − , a + 1 , S c ) then γ and γ are the same.Therefore assume that this is not the case; we can always do this if g ≥ g, n ) = (2 , γ − γ lies in the kernel of the map φ from (40) follows from the fact that λ vanishes on M ,k and λ is equivalentto δ irr on M ,k , for all k for which these spaces are defined. Thus λ · γ and λ · γ are both equal to δ A . Here A is the graph with two genus 0vertices connected by b edges: one has marked points indexed by S and a + 1self-loops, and the other has marked points indexed by S c and c self-loops.Suppose that γ and γ are algebraically equivalent in M λ g,n . The restric-tion sequence(29) R ( M g,n \ M λ i g,n ) −→ R g ( M g,n ) −→ R g ( M λ i g,n ) −→ γ and γ to boundarystrata in M g,n ,(30) γ − γ ∈ R ( M g,n \ M λ g,n ) . However R ( M g,n \ M λ g,n ) is generated by one element δ B , where B is thegraph with a unique vertex of genus 0, g self loops and n half edges.Set K := 2 a + b + | S | L := 2 c + b + | S c | − K := 2 c + b + | S c | − L := 2 a + b + | S | − . (32) RENZO CAVALIERI AND STEPHANIE YANG
Γ(0 , , , ∅ ) Γ(0 , , , { } ) A B
Figure 3.
The graphs Γ(0 , , , ∅ ) and Γ(0 , , , { } ) for( g, n ) = (2 , A and B .For i = 1 , γ i := im (cid:0) ι Γ i : M ,K i × M ,L i +3 → M g,n (cid:1) , and(34) δ B := im (cid:0) ι B : M , g + n → M g,n (cid:1) . Note that L and L cannot both be zero for ( g, n ) = (2 , a = 0, b = 1, and c = 1 (recall that a is non-negative while b and c are positive), which implies ( g, n ) = (2 , L = 0, then L = 0 and we have the following equations which followfrom Lemma 1.1 and the fact that κ a restricted to a boundary divisor if thesum of the pull-backs of κ a on each factor of the gluing map. κ K γ = 12 g − ( g − κ κ L γ = 0(35) κ K γ = 0 κ κ L γ = 12 g − ( g − κ K δ B = 12 g g ! κ κ L δ B = 12 g g ! (cid:18)(cid:18) g + n − (cid:19) − (cid:19) (37)These are incompatible with equation (30). If L = 0 and L = 0 a similarargument holds.Now consider the final case where both L and L are nonzero. Theequations κ K κ L γ = 124 κ K κ L γ = 0 κ K κ L δ B = (cid:18) g + n − K + 1 (cid:19) − κ g − n γ = 0 κ g − n γ = 0 κ g − n δ B = 12 g g !(39)show the independence of the strata γ − γ and δ B . Thus γ cannot bealgebraically equivalent to γ in M λ g,n . (cid:3) Corollary 5.
The dimension of the kernel of the map (40) φ : R g ( M λ g,n ) → hom (cid:16) R g − n ( M λ g,n ) , R g − n ( M λ g,n ) (cid:17) . AUTOLOGICAL PAIRINGS ON MODULI SPACES OF CURVES 9 γ :
112 34 −
11 2 3 4 γ : − η : −
11 2 3 4 η :
11 2 3 4 − Figure 4.
The classes γ , γ , η , and η for ( g, n ) = (4 , goes to infinity as g or n go to infinity.Proof. We exhibit a set of roughly g + n/ φ . Let n denote the set { , . . . , n } . Set(41) γ i = δ (0 , , g − , i ) − δ ( g − , , , n \ i )for i = 1 , . . . , ⌊ n/ ⌋ , and(42) η j = δ ( j, , g − j − , n ) − δ ( g − j − , , j + 1 , ∅ )for j = 1 , . . . , g −
2. Since λ γ i = λ η j = 0, the classes lie in the kernel of φ .The equations ψ i +11 κ g + n − − i γ j = A i δ ij (43) ψ i +11 κ g + n − − i η j = 0(44) ψ n +2 i +11 κ g − − i γ j = 0(45) ψ n +2 i +11 κ g − − i η j = B i δ ij (46)(here δ ij is Kronecker’s delta - not a boundary stratum, and A i , B i arenonzero real numbers) show that these classes are independent. Modulo R ( M g,n \ M λ g,n ), which is one-dimensional, roughly g + ⌊ n/ ⌋ − (cid:3) Failure of Poincar´e duality for M λ i g,n We first outline the strategy of proof for Proposition 3. Fix g and i ≤ g −
1. A generalization of the construction in § S m ⊆ R g ( M g, ( i +1)+ m ) in the annihilator of λ i . The first problem in showingthat at least one such class is nonzero in R g ( M λ i g, ( i +1)+ m ) is that the kernelof the restriction sequence (6) is not known to be tautological. This is adifficult question that we cannot tackle at present. We therefore assumesuch kernel to be tautological. Even so, the dimension of R i − ( M g,n \ M λ i g,n )grows quickly as g or n increase. For a fixed g we bound the order ofgrowth by i n . By proving that S m spans a linear subspace of dimension( i + 1) m , we conclude that eventually some classes in S m will be nonzero in R g ( M λ i g, ( i +1)+ m ). · · · i − i i + 1 — · · · i − i i + 1 Figure 5.
The class σ ∈ R g ( M λ i g,i +1 ) Proof of Proposition 3.
Let σ ∈ R g ( M g,i +1 ) be the difference of boundaryclasses illustrated in Figure 5. Intersecting either of the two strata with λ i results in 1 / i times the class of the graph where all genus one vertices arereplaced with loops, and thus σλ i = 0. We set S = { σ } .For m >
0, and a = ( a , . . . , a m ) an m -tuple of numbers between 1 and i + 1, let σ a ∈ R g ( M g, ( i +1)+ m ) be the class obtained by decorating bothgraphs of σ with the j -th mark on the a j -th vertex for j = 1 , . . . , m . Theset S m of all possible such classes in M λ i g, ( i +1)+ m has cardinality ( i + 1) m .We construct inductively a set T m of classes in complementary codimen-sion which is dual to S m . Our base case is m = 0, where the vector τ canbe chosen to be a scalar multiple of an appropriate product of ψ classes.Consider the universal family(47) π : M g, ( i +1)+ m +1 → M g, ( i +1)+ m , and note that(48) π ∗ σ a = i +1 X k =1 σ ( a ,k ) . where ( a , k ) is the sequence with k appended to the end of a . If D j,m +1 denotes the divisor image of the j -th section in M g, ( i +1)+ m +1 , then for any m -tuple a ,(49) D j,m +1 σ ( a ,k ) = 0if j = k . By the projection formula, and equations (48) and (49),(50) D j,m +1 π ∗ ( τ b ) σ ( a ,k ) equals the class of a point if j = k and a = b , and vanishes otherwise.Therefore the set:(51) T m +1 := { π ∗ τ b ,...,b m D b m +1 ,m +1 } b gives a dual basis to S m +1 .The growth of dim R i − ( M g,n \ M λ i g,n ) with respect to n is at most O ( i n ).To see this, note that the decorated dual graph of any class in R i − ( M g,n \ M λ i g,n )has at most i vertices, and e edges, where g − i + 1 ≤ e ≤ g . The totalnumber of possibly unstable graphs without marked points satisfying theseconditions is independent of n . AUTOLOGICAL PAIRINGS ON MODULI SPACES OF CURVES 11
Graphs with exactly i vertices have g edges, and hence classes supportedon these strata are pure boundary. For a given graph there are i n possibleways of distributing the marked points on the vertices.For graphs with strictly less than i vertices, there are at most ( i − n ways to distribute the marks on the vertices. Each vertex can be decoratedwith a monomial in ψ and κ classes [GP, Proposition 11], of degree ≤ i − κ classes to decoratethe vertices is independent of n . The number of monomials in ψ classes ofdegree ≤ i − n , yielding an order of O (( i − n n k )
Is the map (52) φ : R j ( M λ i g,n ) → hom (cid:16) R g − n − i − j ( M λ i g,n ) , R g − n − i ( M λ i g,n ) (cid:17) injective for j ≤ g − ? References [AC] Enrico Arbarello and Maurizio Cornalba,
Combinatorial and algebro-geometriccohomology classes on the moduli spaces of curves , J. Algebraic Geom. (1996),no. 4, 705–749. MR A conjectural description of the tautological ring of the moduli spaceof curves , Moduli of curves and abelian varieties, Aspects Math., E33, Vieweg,Braunschweig, 1999, pp. 109–129. MR
Hodge integrals and Gromov-Witten the-ory , Invent. Math. (2000), no. 1, 173–199. MR
Logarithmic series and Hodge integrals in the tautological ring , Michi-gan Math. J. (2000), 215–252. With an appendix by Don Zagier; Dedi-cated to William Fulton on the occasion of his 60th birthday. MR Relative maps and tautological classes , J. Eur. Math. Soc. (JEMS) (2005), no. 1, 13–49. MR Intersection theory , 2nd ed., Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Resultsin Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys inMathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR
Constructions of nontautological classeson moduli spaces of curves , Michigan Math. J. (2003), no. 1, 93–109. MR On the tautological ring of M g,n , Turkish J. Math. (2001), no. 1, 237–243. MR Relative virtual localization and vanishing of tautological classes on mod-uli spaces of curves , Duke Math. J. (2005), no. 1, 1–37. MR
Moduli of curves , Graduate Texts in Mathematics,vol. 187, Springer-Verlag, New York, 1998. MR + ] Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, RichardThomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow, Mirror symmetry , ClayMathematics Monographs, vol. 1, American Mathematical Society, Providence,RI, 2003. With a preface by Vafa. MR
Intersection theory of moduli space of stable n -pointed curves ofgenus zero , Trans. Amer. Math. Soc. (1992), no. 2, 545–574. MR Notes on psi classes . available for download at .[L] Eduard Looijenga,
On the tautological ring of M g , Invent. Math. (1995),no. 2, 411–419. MR Towards an enumerative geometry of the moduli space of curves ,Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkh¨auser Boston,Boston, MA, 1983, pp. 271–328. MR On C -complexes , Ann. of Math. (2) (1940), 809–824.MR 0002545 (2,73d)[Y] Stephanie Yang, Intersection numbers on M g,m . Preprint. AUTOLOGICAL PAIRINGS ON MODULI SPACES OF CURVES 13
Colorado State University, Department of Mathematics, Weber Building,Fort Collins, CO 80523-1874
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