Double ramification cycles on the moduli spaces of curves

F. Janda; R. Pandharipande; A. Pixton; D. Zvonkine

doi:10.1007/s10240-017-0088-x

Double ramification cycles on the moduli spaces of curves

Janda, F.; Pandharipande, R.; Pixton, A.; Zvonkine, D. 2017-05-10 00:00:00 DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES by F. JANDA, R. PANDHARIPANDE, A. PIXTON, and D. ZVONKINE ABSTRACT 1 1 1 Curves of genus g which admit a map to P with speciﬁed ramiﬁcation proﬁle μ over 0 ∈ P and ν over ∞∈ P deﬁne a double ramiﬁcation cycle DR (μ, ν) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramiﬁcation cycles on the moduli space of Deligne-Mumford stable curves. The cycle DR (μ, ν) for stable curves is deﬁned via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for DR (μ, ν) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramiﬁcation cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case. When μ = ν =∅, the formula for double ramiﬁcation cycles expresses the top Chern class λ of the Hodge bundle of M as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given. 0. Introduction 0.1. Tautological rings Let M be the moduli space of stable curves of genus g with n marked points. g,n SinceMumford’s article[28], there has been substantial progress in the study of the structure of the tautological rings ∗ ∗ R (M ) ⊂ A (M ) g,n g,n of the moduli spaces of curves. We refer the reader to [12]for asurvey of thebasic deﬁnitions and properties of R (M ). g,n The recent results [21, 31, 32] concerning relations in R (M ), conjectured to g,n be all relations [33], may be viewed as parallel to the presentation A Gr(r, n) = Q c (S), ..., c (S) / s (S), ..., s (S) 1 r n−r+1 n of the Chow ring of the Grassmannian via the Chern and Segre classes of the universal subbundle S → Gr(r, n). The Schubert calculus for the Grassmannian contains several explicit formulas for geo- metric loci. Our main result here is an explicit formula for the double ramiﬁcation cycle in R (M ). g,n All Chow groups will be taken with Q-coefﬁcients. DOI 10.1007/s10240-017-0088-x 222 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE 0.2. Double ramiﬁcation cycles 0.2.1. Notation Double ramiﬁcation data for maps will be speciﬁed by a vector A = (a ,..., a ), a ∈ Z 1 n i satisfying the balancing condition a = 0. i=1 The integers a are the parts of A. We separate the positive, negative, and 0 parts of A as follows. The positive parts of A deﬁne a partition μ = μ + ··· + μ . 1 (μ) The negatives of the negative parts of A deﬁne a second partition ν = ν + ··· + ν . 1 (ν) Up to a reordering of the parts, we have (1)A = (μ ,...,μ , −ν ,..., −ν , 0,..., 0 ). 1 (μ) 1 (ν) n−(μ)−(ν) Since the parts of A sum to 0, the partitions μ and ν must be of the same size. Let D =|μ|=|ν| be the degree of A. Let I be the set of markings corresponding to the 0 parts of A. The various constituents of A are permitted to be empty. The degree 0 case occurs when μ = ν =∅. If I =∅,then n = (μ) + (ν). The empty vector A is permitted, then n = 0, D = 0, and μ = ν = I =∅. There will be no disadvantage in assuming A is ordered as in (1). DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 223 0.2.2. Nonsingular curves Let M ⊂ M be the moduli space of nonsingular pointed curves. Let g,n g,n A = (a ,..., a ) 1 n be a vector of double ramiﬁcation data as deﬁned in Section 0.2.1.Let Z (A) ⊂ M g g,n be the locus parameterizing curves [C, p ,..., p ]∈ M satisfying 1 n g,n (2) O a p = O . C i i C Condition (2) is algebraic and deﬁnes Z (A) canonically as a substack of M . g g,n If [C, p ,..., p ]∈ Z (A), the deﬁning condition (2) yields a rational function (up 1 n g to C -scaling), (3) f : C → P , 1 1 of degree D with ramiﬁcation proﬁle μ over 0 ∈ P and ν over ∞∈ P . The markings corresponding to 0 parts lie over P \{0, ∞}. Conversely, every such morphism (3), up to C -scaling, determines an element of Z (A). We may therefore view Z (A) ⊂ M as the moduli space of degree D maps (up g g,n to C -scaling), f : C → P , with ramiﬁcation proﬁles μ and ν over 0 and ∞ respectively. The term double ramiﬁcation is motivated by the geometry of the map f . The dimension of Z (A) is easily calculated via the theory of Hurwitz covers. Ev- ery map (3) can be deformed within Z (A) to a map with only simple ramiﬁcation over 1 1 P \{0, ∞}. The number of simple branch points in P \{0, ∞} is determined by the Riemann-Hurwitz formula to equal (μ) + (ν) + 2g − 2. The dimension of the irreducible components of Z (A) can be calculated by varying the branch points [36] to equal (μ) + (ν) + 2g − 2 + (I) − 1 = 2g − 3 + n. The −1 on the left is the effect of the C -scaling. Hence, Z (A) is of pure codimension g in M . g,n When considering Z (A),we always assume the stability condition 2g − 2 + n > 0holds. g 224 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE 0.2.3. The Abel-Jacobi map ct Let M ⊂ M be the moduli space of curves of compact type. The universal g,n g,n Jacobian ct Jac → M g,n g,n is the moduli space of line bundles on the universal curve of degree 0 on every irreducible component of every ﬁber. There is a natural compactiﬁcation of Z (A) in the moduli space of compact type curves, ct ct Z (A) ⊂ Z (A) ⊂ M , g g,n obtained from the geometry of the universal Jacobian. Consider the universal curve over the compact type moduli space ct ct ct π : C → M . g,n ct The double ramiﬁcation vector A = (a ,..., a ) determines a line bundle L on C of 1 n relative degree 0, L = a [p ], i i ct ct where p ⊂ C is the section of π corresponding to the marking p . i i ct By twisting L by components of the π inverse images of the boundary divisors of ct M , we can easily construct a line bundle L which has degree 0 on every component g,n ct ct of every ﬁber of π . The result L is unique up to twisting by the π inverse images of ct the boundary divisors of M . g,n Via L , we obtain a section of the universal Jacobian, ct (4) φ : M → Jac . g,n g,n ct Certainly the closure of Z (A) in M lies in the φ inverse image of the 0-section S ⊂ g,n Jac of the relative Jacobian. We deﬁne g,n ct −1 ct (5) Z (A) = φ (S) ⊂ M . g g,n ct The substack Z (A) is independent of the choice of L . From the deﬁnitions, we see ct Z (A) ∩ M = Z (A). g,n g g DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 225 ct While S ⊂ Jac is of pure codimension g, Z (A) typically has components of g,n excess dimension (obtained from genus 0 components of the domain). Acycle classof the expected dimension vir ct ct g ct DR (A) = Z (A) ∈ A M g g g,n ∗ ct is deﬁned by φ ([S]). A closed formula for DR (A) was obtained by Hain [19]. A simpler approach was later provided by Grushevsky and Zakharov [18]. 0.2.4. Stable maps to rubber Let A = (a ,..., a ) be a vector of double ramiﬁcation data. The moduli space 1 n M P ,μ,ν g,I parameterizes stable relative maps of connected curves of genus g to rubber with ramiﬁcation proﬁles μ, ν over the point 0, ∞∈ P respectively. The tilde indicates a rubber target. We refer the reader to [25, 26] for the foundations. There is a natural morphism : M P ,μ,ν → M g,I g,n forgetting everything except the marked domain curve. Let R/{0, ∞} be a rubber target, and let f : C → R/{0, ∞} ∈ M P ,μ,ν g,I be a moduli point. If C is of compact type and R = P ,then ct [ f ] ∈ Z (A) ⊂ M . g,n Hence, we have the inclusion ct Z (A) ⊂ Im() ⊂ M . g,n 1 ∼ The virtual dimension of M (P ,μ,ν) is 2g − 3 + n.Wedeﬁne the double rami- g,I ﬁcation cycle to be the push-forward ∼ vir 1 g DR (A) = M P ,μ,ν ∈ A (M ). g ∗ g,I g,n 4 ct Hence, Z (A) ⊂ Z (A) is not always dense. 5 1 R is a chain of P s. 226 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE 0.2.5. Basic properties The ﬁrst property of DR (A) is a compatibility with the Abel-Jacobi construction of Section 0.2.3.Let ct ι : M → M g,n g,n be the open inclusion map. Proposition 1 (Marcus-Wise [27]). — We have ∗ ct g ct ι DR (A) = DR (A) ∈ A M . g g,n By deﬁnition, DR (A) is a Chow class on the moduli space of stable curves. In fact, the double ramiﬁcation cycle lies in the tautological ring. Proposition 2 (Faber-Pandharipande [11]). — DR (A) ∈ R (M ). g g,n The proof of [11] provides an algorithm to calculate DR (A) in the tautological ring, but the complexity of the method is too great: there is no apparent way to obtain an explicit formula for DR (A) directly from [11]. 0.3. Stable graphs and strata 0.3.1. Summation over stable graphs The strata of M are the quasi-projective subvarieties parameterizing pointed g,n curves of a ﬁxed topological type. The moduli space M is a disjoint union of ﬁnitely many g,n strata. The main result of the paper is a proof of an explicit formula conjectured by Pixton [34] in 2014 for DR (A) in the tautological ring R (M ). The formula is written in g g,n terms of a summation over stable graphs indexing the strata of M .Wereviewhere g,n the standard indexing of the strata of M by stable graphs. g,n 0.3.2. Stable graphs The strata of the moduli space of curves correspond to stable graphs = (V, H, L, g : V → Z ,v : H → V,ι : H → H) ≥0 satisfying the following properties: Compatibility on the smaller open set of curves with rational tails was proven earlier in [4]. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 227 (i) V is a vertex set with a genus function g : V → Z , ≥0 (ii) H is a half-edge set equipped with a vertex assignment v : H → Vand an involution ι, (iii) E, the edge set, is deﬁned by the 2-cycles of ι in H (self-edges at vertices are permitted), (iv) L, the set of legs, is deﬁned by the ﬁxed points of ι and is placed in bijective correspondence with a set of markings, (v) the pair (V, E) deﬁnes a connected graph, (vi) for each vertex v, the stability condition holds: 2g(v) − 2 + n(v) > 0, where n(v) is the valence of at v including both half-edges and legs. An automorphism of consists of automorphisms of the sets V and H which leave in- variant the structures L, g, v,and ι.Let Aut( ) denote the automorphism group of . The genus of a stable graph is deﬁned by: g( ) = g(v) + h ( ). v∈V A quasi-projective stratum of M corresponding to Deligne-Mumford stable curves of g,n ﬁxed topological type naturally determines a stable graph of genus g with n legs by con- sidering the dual graph of a generic pointed curve parameterized by the stratum. Let G be the set of isomorphism classes of stable graphs of genus g with n legs. g,n The set G is ﬁnite. g,n 0.3.3. Strata classes To each stable graph ∈ G , we associate the moduli space g,n M = M . g(v),n(v) v∈V There is a canonical morphism (6) ξ : M → M g,n with image equal to the closure of the stratum associated to the graph . To construct ξ , a family of stable pointed curves over M is required. Such a family is easily deﬁned by attaching the pull-backs of the universal families over each of the M along the g(v),n(v) sections corresponding to the two halves of each edge in E( ). The degree of ξ is |Aut( )|. 228 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Each half-edge h ∈ H( ) determines a cotangent line L → M . If h ∈ L( ),then L is the pull-back via ξ of the corresponding cotangent line of M . h g,n If h is a side of an edge e ∈ E( ),then L is the cotangent line of the corresponding side of a node. Let ψ = c (L ) ∈ A (M , Q). h 1 h 0.4. Pixton’s formula 0.4.1. Weightings Let A = (a ,..., a ) be a vector of double ramiﬁcation data. Let ∈ G be a 1 n g,n stable graph of genus g with n legs. A weighting of is a function on the set of half-edges, w : H( ) → Z, which satisﬁes the following three properties: (i) ∀h ∈ L( ), corresponding to the marking i ∈{1,..., n}, w(h ) = a , i i (ii) ∀e ∈ E( ), corresponding to two half-edges h, h ∈ H( ), w(h) + w h = 0, (iii) ∀v ∈ V( ), w(h) = 0, v(h)=v where the sum is taken over all n(v) half-edges incident to v. If the graph has cycles, may carry inﬁnitely many weightings. Let r be a positive integer. A weighting mod r of is a function, w : H( ) →{0,..., r − 1}, which satisﬁes exactly properties (i)–(iii) above, but with the equalities replaced, in each case, by the condition of congruence mod r. For example, for (i), we require w(h ) = a mod r. i i h ( ) Let W be the set of weightings mod r of . The set W is ﬁnite, with cardinality r . ,r ,r We view r as a regularization parameter. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 229 0.4.2. Pixton’s conjecture Let A = (a ,..., a ) be a vector of double ramiﬁcation data. Let r be a positive 1 n d,r d integer. We denote by P (A) ∈ R (M ) the degree d component of the tautological g,n class 1 1 ξ exp a ψ ∗ h 1 i i h ( ) |Aut( )| r ∈G w∈W i=1 g,n ,r 1 − exp(−w(h)w(h )(ψ + ψ )) h h × , ψ + ψ h h e=(h,h )∈E( ) in R (M ). g,n Inside the push-forward in the above formula, the ﬁrst product exp a ψ i i i=1 is over h ∈ L( ) via the correspondence of legs and markings. The second product is over all e ∈ E( ).The factor 1 − exp(−w(h)w(h )(ψ + ψ )) h h ψ + ψ h h is well-deﬁned since • the denominator formally divides the numerator, • the factor is symmetric in h and h . No edge orientation is necessary. d,r The following fundamental polynomiality property of P (A) has been proven by Pixton. Proposition 3 (Pixton [35]). — For ﬁxed g, A, and d , the class d,r d P (A) ∈ R (M ) g,n is polynomial in r (for all sufﬁciently large r). d d,r We denote by P (A) the value at r = 0 of the polynomial associated to P (A) by g g Proposition 3.Inother words, P (A) is the constant term of the associated polynomial in r. For the reader’s convenience, Pixton’s proof of Proposition 3 is given in the Appendix. The main result of the paper is a proof of the formula for double ramiﬁcation cycles conjectured earlier by Pixton [34]. 230 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Theorem 1. —For g ≥ 0 and double ramiﬁcation data A,wehave −g g g DR (A) = 2 P (A) ∈ R (M ). g g,n For d < g, the classes P (A) do not yet have a geometric interpretation. However for d > g, the following vanishing conjectured by Pixton is now established. Theorem 2 (Clader-Janda [8]). — For all g ≥ 0, double ramiﬁcation data A,and d > g, we have d d P (A) = 0 ∈ R (M ). g,n Pixton [34] has further proposed a twist of the formula for P (A) by k ∈ Z.The codimension g class in the k = 1 case has been (conjecturally) related to the moduli spaces of meromorphic differential in the Appendix of [13]. The vanishing result of Clader-Janda [8] proves Pixton’s vanishing conjecture in codimensions d > g for all k. The k-twisted theory will be discussed in Section 1. In a forthcoming paper [22], we will generalize Pixton’s formula to the situation of maps to a P -rubber bundle over a target manifold X. 0.5. Basic examples 0.5.1. Genus 0 Let g = 0and A = (a ,..., a ) be a vector of double ramiﬁcation data. The graphs 1 n ∈ G are trees. Each ∈ G has a unique weighting mod r (for each r). We can directly 0,n 0,n calculate: d 2 2 (7)P (A) = P (A) = exp a ψ − a δ , 0 i I,J 0 i I d≥0 i=1 I J={1,...,n} |I|,|J|≥2 where a = a . I i i∈I By Theorem 1, the double ramiﬁcation cycle is the degree 0 term of (7), −0 0 DR (A) = 2 P (A) = 1. The vanishing of Theorem 2 implies 2 2 1 a ψ − a δ = 0 ∈ R (M ), i I,J 0,n i I i=1 I J={1,...,n} |I|,|J|≥2 where δ ∈ R (M ) is the class of the boundary divisor with marking distribution I J. I,J 0,n The above vanishing can be proven here directly by expressing ψ classes in terms of boundary divisors. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 231 0.5.2. Genus 1 Let g = 1and A = (a ,..., a ) be a vector of double ramiﬁcation data. Denote by 1 n δ ∈ R (M ) the boundary divisor of singular stable curves with a nonseparating node. 0 1,n Our convention is δ = ξ [M ],ξ : M → M . 0 ∗ 0,n+2 0,n+2 1,n The division by the order of the automorphism group is included in the deﬁnition of δ . Denote by δ ∈ R (M ) I 1,n the class of the boundary divisor of curves with a rational component carrying the mark- ings I ⊂{1,..., n} and an elliptic component carrying the markings I . We can calculate directly from the deﬁnitions: 1 2 2 (8)P (A) = a ψ − a δ − δ . i I 0 1 i I i=1 I⊂{1,...,n} |I|≥2 As before, a = a .ByTheorem 1, I i i∈I DR (A) = P (A). In genus 1, the virtual class plays essentially no role (since the moduli spaces of maps to rubber are of expected dimension). The genus 1 formula was already known to Hain [19]. It is instructive to compute the coefﬁcient − of δ .The class δ corresponds to 0 0 the graph with one vertex of genus 0 and one loop, h ( ) = 1. According to the deﬁnition, the coefﬁcient of δ is the r-free term of the polynomial r−1 w(r − w) , w=1 which is −B =− . 6 232 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE 0.5.3. Degree 0 Let g ≥ 0and A = (0,..., 0),so μ = ν =∅. The canonical map from the moduli of stable maps to the moduli of curves is then an isomorphism ∼ ∼ : M P , ∅, ∅ −→ M . g,n g,n By an analysis of the obstruction theory, g g DR (0,..., 0) = (−1) λ ∈ R (M ) g g g,n where λ is the top Chern class of the Hodge bundle E → M . g,n Let ∈ G be a stable graph. By the deﬁnition of a weighting mod r and the condi- g,n tion a = 0 for all markings i, the weights w(h), w h on the two halves of every separating edge e of must both be 0. The factor in Pixton’s formula for e, 1 − exp(−w(h)w(h )(ψ + ψ )) h h ψ + ψ h h then vanishes and kills the contribution of to P (0,..., 0). Let g ≥ 1. Let ξ be the map to M associated to the divisor of curves with non- g,n separating nodes ξ : M → M . g−1,n+2 g,n Since graphs with separating edges do not contribute to P (0,..., 0) and the trivial graph with no edges does not contribute in codimension g, g−1 P (0,..., 0) = ξ (0,..., 0) g g−1 for an explicit tautological class g−1 g−1 (0,..., 0) ∈ R (M ). g−1,n+2 g−1 Corollary 3. —For g ≥ 1, we have g−1 −g g λ = (−2) ξ (0,..., 0) ∈ R (M ). g ∗ g,n g−1 DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 233 ct The class λ is easily proven to vanish on M via the Abel-Jacobi map (4)and g,n well-known properties of the moduli space of abelian varieties [38]. Hence, there exists a g−1 Chow class γ ∈ A (M ) satisfying g−1,n+2 g−1,n+2 λ = ξ γ ∈ A (M ). g ∗ g−1,n+2 g,n g−1 Corollary 3 is a much stronger statement since (0,..., 0) is a tautological class on g−1 M given by an explicit formula. No such expressions were known before. g−1,n+2 • For M , Corollary 3 yields 1,n 1−1 1 λ =− ξ (0,..., 0) ∈ R (M ). 1 ∗ 1,n 1−1 Moreover, by calculation (8), 1−1 (0,..., 0) =− [M ], 0,n+2 1−1 since δ already carries a for the graph automorphism. Hence, we recover the well- known formula λ = ξ [M ]∈ R (M ). 1 ∗ 0,n+2 1,n • For M , Corollary 3 yields 2−1 2 λ = ξ (∅) ∈ R (M ). 2 ∗ 2 2−1 Denote by α the class of the boundary stratum of genus 0 curves with two self- intersections. We include the division by the order of the automorphism group in the deﬁnition, α = ξ [M ]. ∗ 0,4 Denote by β the class of the boundary divisor of genus 1 curves with one self-intersection multiplied by the ψ -class at one of the branches. Then we have 1 1 2−1 (9) ξ (∅) = α + β. 2−1 36 60 The classes α and β form a basis of R (M ). The expression 1 1 λ = α + β 144 240 can be checked by intersection with the two boundary divisors of M . 2 234 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE The coefﬁcient of α in (9) is obtained from the r-free term of w w (r − w )(r − w ). 1 2 1 2 1≤w ,w ≤r−1 1 2 The answer is B = . The coefﬁcient of β is the r-free term of r−1 2 2 − w (r − w) 2r w=1 1 1 given by − B = . 2 60 0.6. Strategy of proof The proof of Proposition 2 given in [11] uses the Gromov-Witten theory of P relative to the point ∞∈ P . The localization relations of [11] intertwine Hodge classes on the moduli spaces of curves with cycles obtained from maps to rubber. The interplay is very complicated with many auxiliary classes (not just the double ramiﬁcation cycles). Our proof of Theorem 1 is obtained by studying the Gromov-Witten theory of 1 8 1 1 1 P with an orbifold point BZ at 0 ∈ P and a relative point ∞∈ P .Let (P [r], ∞) denote the resulting orbifold/relative geometry. The proof may be explained conceptu- ally as studying the localization relations for all (P [r], ∞) simultaneously.For large r,the localization relations (after appropriate rescaling) may be viewed as having polynomial coefﬁcients in r. Then the r = 0 specialization of the linear algebra exactly yields Theo- rem 1. The regularization parameter r in the deﬁnition of Pixton’s formula in Section 0.4 is related to the orbifold structure BZ . Our proof is inspired by the structure of Pixton’s formula—especially the existence of the regularization parameter r. The argument requires two main steps: • Let M be the moduli space of rth roots (with respect to the tensor product) of g;a ,...,a 1 n the bundles O ( a x ) associated to pointed curves [C, p ,..., p ].Let C i i 1 n i=1 π : C → M g;a ,...,a g;a ,...,a 1 n 1 n be the universal curve, and let L be the universal rth root over the universal curve. Chiodo’s formulas [6] allow us to prove that the push-forward of r · c (−R π L) to M g ∗ g,n −g g,r and 2 P (A) are polynomials in r with the same constant term. Pixton’s formula there- fore has a geometric interpretation in terms of the intersection theory of the moduli space of rth roots. 8 Z BZ is the quotient stack obtained by the trivial action of Z on a point. r r = rZ 9 1 The target (P [r], ∞) was previously studied in [24] in the context of Hurwitz-Hodge integrals. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 235 • We use the localization formula [15] for the virtual class of the moduli space of stable maps to the orbifold/relative geometry (P [r], ∞). The monodromy conditions at 0 are given by μ and the ramiﬁcation proﬁle over ∞ is given by ν . The push-forward of the localization formula to M is a Laurent series in the equivariant parameter t and in r. g,n −1 0 The coefﬁcient of t r must vanish by geometric considerations. We prove the relation −1 0 obtained from the coefﬁcient of t r has only two terms. The ﬁrst is the constant term in r of the push-forward of r · c (−R π L) to M . The second term is the double ramiﬁ- g ∗ g,n cation cycle DR (A) with a minus sign. The vanishing of the sum of the two terms yields Theorem 1. 1. Moduli of stable maps to BZ 1.1. Twisting by k Let k ∈ Z.Avector A = (a ,..., a ) of k-twisted double ramiﬁcation data for genus g ≥ 0 1 n is deﬁned by the condition (10) a = k(2g − 2 + n). i=1 For k = 0, the condition (10)does not depend upon on the genus g and specializes to the balancing condition of the double ramiﬁcation data of Section 0.2.1. d,k d We recall the deﬁnition [34]ofPixton’s k-twisted cycle P (A) ∈ R (M ).Incase g,n k = 0, d,0 d d P (A) = P (A) ∈ R (M ). g,n g g Let ∈ G be a stable graph of genus g with n legs. A k-weighting mod r of is a g,n function on the set of half-edges, w : H( ) →{0,..., r − 1}, which satisﬁes the following three properties: (i) ∀h ∈ L( ), corresponding to the marking i ∈{1,..., n}, w(h ) = a mod r, i i (ii) ∀e ∈ E( ), corresponding to two half-edges h, h ∈ H( ), w(h) + w h =0mod r We always assume the stability condition 2g − 2 + n > 0holds. 236 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE (iii) ∀v ∈ V( ), w(h) = k 2g(v) − 2 + n(v) mod r, v(h)=v where the sum is taken over all n(v) half-edges incident to v. We denote by W the set of all k-weightings mod r of . ,r,k Let A be a vector of k-twisted ramiﬁcation data for genus g. For each positive d,r,k d integer r,let P (A) ∈ R (M ) be the degree d component of the tautological class g,n 1 1 2 2 −k κ (v) a ψ 1 h ξ e e h ( ) |Aut( )| r ∈G w∈W v∈V( ) i=1 g,n ,r,k −w(h)w(h )(ψ +ψ ) 1 − e × , ψ + ψ h h e=(h,h )∈E( ) in R (M ). g,n All the conventions here are the same as in the deﬁnitions of Section 0.4.2.Anew factor appears: κ (v) is the ﬁrst κ class, κ (v) ∈ R (M ), 1 g(v),n(v) on the moduli space corresponding to a vertex v of the graph . As in the untwisted case, polynomiality holds (see the Appendix). Proposition 3 (Pixton [35]). — For ﬁxed g, A, k, and d , the class d,r,k d P (A) ∈ R (M ) g,n is polynomial in r (for all sufﬁciently large r). d,k d,r,k We denote by P (A) the value at r = 0 of the polynomial associated to P (A) g g d,k by Proposition 3 .Inother words, P (A) is the constant term of the associated polynomial in r. In case k = 0, Proposition 3 specializes to Proposition 3. Pixton’s proof is given in the Appendix. i+1 11 i Our convention is κ = π (ψ ) ∈ R (M ) where i ∗ g(v),n(v) n(v)+1 π : M → M g(v),n(v)+1 g(v),n(v) is the map forgetting the marking n(v) + 1. For a review of κ and cotangent ψ classes, see [16]. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 237 1.2. Generalized r-spin structures 1.2.1. Overview Let [C, p ,..., p ]∈ M be a nonsingular curve with distinct markings and 1 n g,n canonical bundle ω .Let ω = ω (p + ··· + p ) log C 1 n be the log canonical bundle. Let k and a ,..., a be integers for which k(2g − 2 + n) − a is divisible by a 1 n i positive integer r.Then rth roots L of the line bundle ⊗k ω − a p i i log on C exist.The spaceofsuch rth tensor roots possesses a natural compactiﬁcation r,k M constructed in [5, 23]. The moduli space of rth roots carries a universal curve g;a ,...,a 1 n r,k r,k π : C → M g;a ,...,a g;a ,...,a 1 n 1 n and a universal line bundle r,k L → C g;a ,...,a 1 n equipped with an isomorphism ⊗r ⊗k (11) L −→ ω − a x i i log r,0 r,k on C .Incase k = 0, M is the space of stable maps [1] to the orbifold BZ . g;a ,...,a g;a ,...,a 1 n 1 n r,k We review here the basic properties of the moduli spaces M which we re- g;a ,...,a 1 n quire and refer the reader to [5, 23] for a foundational development of the theory. 1.2.2. Covering the moduli of curves The forgetful map to the moduli of curves r,k : M → M g,n g;a ,...,a 1 n 2g is an r -sheeted covering ramiﬁed over the boundary divisors. The degree of equals 2g−1 r because each rth root L has a Z symmetry group obtained from multiplication by rth roots of unity in the ﬁbers. The Z -action respects the isomorphism (11). r 238 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE To transform to an unramiﬁed covering in the orbifold sense, the orbifold structure of M must be altered. We introduce an extra Z stabilizer for each node of each stable g,n r curve, see [5]. The new orbifold thus obtained is called the moduli space of r-stable curves. 1.2.3. Boundary strata The rth roots of the trivial line bundle over the cylinder are classiﬁed by a pair of integers (12)0 ≤ a , a ≤ r − 1, a + a =0mod r attached to the borders of the cylinder. Therefore, the nodes of stable curves in the com- r,k pactiﬁcation M carry the data (12)with a and a assigned to the two branches g;a ,...,a 1 n meeting at the node. In case thenodeis separating,the data (12) are determined uniquely by the way the genus and the marked points are distributed over the two components of the curve. In case thenodeis nonseparating, the data can be arbitrary and describes different boundary divisors. r,k More generally, the boundary strata of M are described by stable graphs g;a ,...,a 1 n ∈ G together with a k-weighting mod r.The data (12) is obtained from the weighting g,n w of the half-edges of . 1.2.4. Normal bundles of boundary divisors The normal bundle of a boundary divisor in M has ﬁrst Chern class g,n − ψ + ψ , where ψ and ψ are the ψ -classes associated to the branches of the node. In the moduli r,k space of r-stable curves and in the space M , the normal bundle to a boundary g;a ,...,a 1 n divisor has ﬁrst Chern class ψ + ψ (13) − , where the classes ψ and ψ are pulled back from M . g,n The two standard boundary maps M → M g−1,n+2 g,n and M × M → M g ,n +1 g ,n +1 g,n 1 1 2 2 More precisely, the normal bundle of the associated gluing morphism ξ . DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 239 are, in the space of rth roots, replaced by diagrams r,k φ r,k nonsep (14) M ←− B −→ M g−1;a ,...,a ,a ,a a ,a g;a ,...,a 1 n 1 n and r,k r,k ϕ j r,k sep (15) M × M ←− B −→ M . g ;a ,...,a ,a g ;a ,a ,...,a a ,a g;a ,...,a 1 1 n 2 n +1 n 1 n 1 1 The maps φ and ϕ are of degree 1, but in general are not isomorphisms, see [7], Sec- tion 2.3. Remark 1. — There are two natural ways to introduce the orbifold structure on the space of rth roots. In the ﬁrst, every node of the curve contributes an extra Z/rZ in the stabilizer. In the second, a node of type (a , a ) contributes an extra Z/gcd(a , r)Z to the stabilizer. The ﬁrst is used in [5], the second in [1]. Here, we follow the ﬁrst convention to avoid gcd factors appearing the computations. In Remark 2, we indicate the changes which must be made in the computations if the second convention is followed. The ﬁnal result is the same (independent of the choice of convention). 1.2.5. Intersecting boundary strata Let ( ,w ) and ( ,w ) be two stable graphs in G equipped with k-weightings A A B B g,n r,k mod r. The intersection of the two corresponding boundary strata of M can be g;a ,...,a 1 n determined as follows. Enumerate all pairs ( , w) of stable graphs with k-weightings mod r whose edges are marked with letters A, B or both in such a way that contracting all edges outside A yields ( ,w ) and contracting all edges outside B yields ( ,w ). A A B B Assign a factor − ψ + ψ to every edge that is marked by both A and B. The sum of boundary strata given by these graphs multiplied by the appropriate combinations of ψ -classes represents the intersec- tion of the boundary components ( ,w ) and ( ,w ). A A B B 1.3. Chiodo’s formula Let g be the genus, and let k ∈ Z and a ,..., a ∈ Z satisfy 1 n k(2g − 2 + n) − a =0mod r for an integer r > 0. See the Appendix of [16] for a detailed discussion in the case of M . g,n 240 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Following the notation of Section 1.2,let r,k r,k π : C → M g;a ,...,a g;a ,...,a 1 n 1 n r,k be the universal curve and L → C the universal rth root. Our aim here is to de- g;a ,...,a 1 n scribe the total Chern class r,k ∗ ∗ c −R π L ∈ A M g;a ,...,a 1 n following the work of A. Chiodo [6]. Denote by B (x) the m-th Bernoulli polynomial, n xt t te B (x) = . n! e − 1 n=0 Proposition 4. — The total Chern class c(−R π L) is equal to B (k/r) 1 m−1 m+1 |E( )| − (−1) κ (v) m≥1 m(m+1) r (ξ ) e ,w ∗ |Aut( )| ∈G w∈W v∈V( ) g,n ,r,k B (a /r) m−1 m+1 i m (−1) ψ m≥1 m(m+1) h × e i=1 B (w(h)/r) m−1 m+1 m m (−1) [(ψ ) −(−ψ ) ] m≥1 h m(m+1) 1 − e ψ + ψ h h e∈E( ) e=(h,h ) r,k in A (M ). g;a ,...,a 1 n Here, ξ is the map of the boundary stratum corresponding to ( , w) to the ,w r,k moduli space M , g;a ,...,a 1 n r,k ξ : M → M . ,w ,w g;a ,...,a 1 n Every factor in the last product is symmetric in h and h since w r − w m+1 B = (−1) B m+1 m+1 r r for m ≥ 1. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 241 Proof. — The proposition follows from Chiodo’s formula [6] for the Chern charac- ters of R π L, k a B ( ) B ( ) m+1 m+1 r r m (16) ch (r, k; a ,..., a ) = κ − ψ m 1 n m (m + 1)! (m + 1)! i=1 r−1 m m B ( ) r (ψ ) − (−ψ ) m+1 + ξ , w∗ 2 (m + 1)! ψ + ψ w=0 and from the universal relation • m • • b c −E = exp (−1) (m − 1)!ch E , E ∈ D . m≥1 |E( )| The factor r in the proposition is due to the factor of r in the last term of Chiodo’s formula. When two such terms corresponding to the same edge are multi- plied, the factor r partly cancels with the factor in the ﬁrst Chern class of the normal bundle − (ψ + ψ ). Thus only a single factor of r per edge remains. ∗ ∗ Corollary 4. — The push-forward c(−R π L) ∈ R (M ) is equal to ∗ ∗ g,n 2g−1−h ( ) B (k/r) r m−1 m+1 − (−1) κ (v) m≥1 m(m+1) ξ e |Aut( )| ∈G w∈W v∈V( ) g,n ,r,k B (a /r) m−1 m+1 i m (−1) ψ m≥1 m(m+1) h × e i=1 B (w(h)/r) m−1 m+1 m m (−1) [(ψ ) −(−ψ ) ] m≥1 h m(m+1) 1 − e × . ψ + ψ h h e∈E( ) e=(h,h ) Proof. — Because the maps φ and ϕ of Equations (14)and (15)havedegree1,the degree of the map r,k (17) : M → M g,n g;a ,...,a 1 n restricted to the stratum associated to ( , w) is equal to the product of degrees over vertices, that is, (2g −1) v∈V( ) r . r 1 The factor in the last term is ,but the accounts for the degree 2 of the boundary map ξ obtained from the 2 2 ordering of the nodes. 242 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE The formula of the corollary is obtained by push-forward from the formula of Proposi- tion 4.The powerof r is given by |E|+ (2g − 1) =|E|−|V|+ 2 g v v v∈V( ) v∈V( ) 1 1 = h ( ) − 1 + 2 g − h ( ) = 2g − 1 − h ( ). 1.4. Reduction modulo r Let g be the genus, and let k ∈ Z and a ,..., a ∈ Z satisfy 1 n k(2g − 2 + n) − a =0mod r for an integer r > 0. Let A = (a ,..., a ) 1 n be the associated vector of k-twisted double ramiﬁcation data. Proposition 3 (Pixton [35]). — Let ∈ G be a ﬁxed stable graph. Let Q be a polynomial g,n with Q-coefﬁcients in variables corresponding to the set of half-edges H( ). Then the sum F(r) = Q w(h ), ...,w(h ) 1 |H( )| w∈W ,r,k h ( ) over all possible k-weightings mod r is (for all sufﬁciently large r) a polynomial in r divisible by r . Proposition 3 implies Proposition 3 (which implies Proposition 3 under the spe- cialization k = 0). Proposition 3 is the basic statement. See the Appendix for the proof. We will require Proposition 3 to prove the following result. Let r,k : M → M g,n g;a ,...,a 1 n be the map to the moduli of curves. Let d ≥ 0 be a degree and let c denote the d -th Chern class. Proposition 5. — The two cycle classes 2d−2g+1 ∗ d −d d,r,k d r c −R π L ∈ R (M ) and 2 P (A) ∈ R (M ) ∗ d ∗ g,n g,n are polynomials in r (for r sufﬁciently large). Moreover, the two polynomials have the same constant term. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 243 Proof. — Choose a stable graph decorated with κ classes on the vertices and ψ classes on half-edges, [ , γ ],γ = M κ(v) · ψ . v∈V( ) h∈H( ) Here, M is a monomial in the κ classes at v and m is a non-negative integer for each h. v h The decorated graph data [ , γ ] represents a tautological class of degree d.Wewill study the coefﬁcient of the decorated graph in the formula for c (−R π L) given by ∗ d ∗ Corollary 4 for r sufﬁciently large. According to Proposition 3 , the coefﬁcient of [ , γ ] in the formula for the push- forward c (−R π L) is a Laurent polynomial in r. Indeed, the formula of Corollary 4 ∗ d ∗ is a combination of a ﬁnite number of sums of the type of Proposition 3 multiplied by positive and negative powers of r. Each contributing sum is obtained by expanding the exponentials in the formula of Corollary 4 and selecting a degree d monomial. To begin with, let us determine the terms of lowest power in r in the Laurent poly- nomial obtained by summation in the formula of Corollary 4.The (m + 1)st Bernoulli polynomial B has degree m + 1. Therefore, in the formula of Corollary 4, every class m+1 m 16 m−1 of degree m,beit κ , ψ or an edge with a class ψ , appears with a power of r 1 17 equal to or larger. If we take a product of q coefﬁcients of expanded exponentials of m+1 Bernoulli polynomials, we will obtain a power of r equal to or larger. Thus the lowest d+q possible power of r is obtained if we take a product of d classes of degree 1 each. Then, the power of r is equal to which is the lowest possible value. 2d After we perform the summation over all k-weightings mod r of the graph ,we h ( ) obtain a polynomial in r divisible by r by Proposition 3 . The lowest possible power h ( )−2d of r after summation becomes r . Finally, formula of Corollary 4 carries a global 2g−1−h ( ) ∗ factor of r . We conclude c (−R π L) is a Laurent polynomial in r with lowest ∗ d ∗ 2g−2d−1 power of r equal to r . In other words, the product 2d−2g+1 ∗ r c −R π L ∗ d ∗ is a polynomial in r. We now identify more precisely the terms of lowest power in r. As we have seen above, the lowest possible power of r in the formula of Corollary 4 is obtained by choosing only the terms that contain a degree 1 class multiplied by the second Bernoulli polyno- mial, w w w 1 B = − + . r r r 6 The edge contributes 1 to the degree of the class. 17 1 The power of is −m − 1. By larger here, we mean larger power (so smaller pole or polynomial terms). m+1 r 244 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Moreover, in the Bernoulli polynomial only the ﬁrst term will contribute to the lowest power of r. After dropping all the terms which do not contribute to the lowest power of r, the formula of Corollary 4 simpliﬁes to 1 n 2g−1−h ( ) 2 r k a ξ exp − κ (v) exp ψ ∗ 1 h 2 2 |Aut( )| 2r 2r ∈G w∈W v∈V( ) i=1 g,n ,r,k w(h) 1 − exp[ (ψ + ψ )] 2 h h 2r × . ψ + ψ h h e=(h,h )∈E( ) We transform w(h) in the last factor to −w(h)w(h ) for symmetry. Indeed, w(h) = w(h) r − w h =−w(h)w h mod r, so the lowest degree in r is not affected. After factoring out the 2r in the denominators, we obtain −h ( ) 2 2 2g−1−2d −d −k κ (v) a ψ 1 h r · 2 · ξ e e |Aut( )| ∈G w∈W v∈V( ) i=1 g,n ,r,k −w(h)w(h )(ψ +ψ ) 1 − e × . ψ + ψ h h e=(h,h )∈E( ) 2d−2g+1 −d d,k,r Multiplication by r then yields exactly 2 P (A) as claimed. 2. Localization analysis 2.1. Overview Let A = (a ,..., a ) be a vector of double ramiﬁcation data as deﬁned in Sec- 1 n tion 0.2.1, a = 0. i=1 From the positive and negative parts of A, we obtain two partitions μ and ν of the same size |μ|=|ν|. The double ramiﬁcation cycle DR (A) ∈ R (M ) g g,n 1 ∼ is deﬁned via the moduli space of stable maps to rubber M (P ,μ,ν) where I corre- g,I sponds to the 0 parts of A. We prove here the claim of Theorem 1, −g g g DR (A) = 2 P (A) ∈ R (M ). g g,n g DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 245 2.2. Target geometry Following the notation of [24], let P [r] be the projective line with an orbifold point BZ at 0 ∈ P .Let M P [r],ν g,I,μ be the moduli space of stable maps to the orbifold/relative pair (P [r], ∞). The moduli space parameterizes connected, semistable, twisted curves C of genus g with I markings together with a map f : C → P 1 1 where P is a destabilization of P [r] over ∞∈ P [r]. We refer the reader to [1, 24] for the deﬁnitions of the moduli space of stable maps 1 1 to (P [r], ∞). The following conditions are required to hold over 0, ∞∈ P [r]: • The stack structure of the domain curve C occurs only at the nodes over 0 ∈ P [r] and the markings corresponding to μ (which must to be mapped to 0 ∈ P [r]). The monodromies associated to the latter markings are speciﬁed by the parts μ of μ. For each part μ ,let i i μ = μ mod r where 0 ≤ μ ≤ r − 1. i i • The map f is ﬁnite over ∞∈ P [r] on the last component of P with ramiﬁcation data given by ν.The (ν) ramiﬁcation points are marked. The map f satisﬁes the ramiﬁcation matching condition over the internal nodes of the destabiliza- tion P. The moduli space M (P [r],ν) has a perfect obstruction theory and a virtual g,I,μ class of dimension (μ) |ν| μ vir (18)dim M P [r],ν = 2g − 2 + n + − , C g,I,μ r r i=1 see [24, Section 1.1]. Recall n is the length of A, n = (μ) + (ν) + (I). Since |μ|=|ν|, the virtual dimension is an integer. We will be most interested in the case where r > |μ|=|ν|. Then, μ = μ for all the parts of μ, and formula (18) for the virtual dimension simpliﬁes to vir dim M P [r],ν = 2g − 2 + n. C g,I,μ 246 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE 2.3. C -ﬁxed loci ∗ 1 The standard C -action on P deﬁned by ξ ·[z , z ]=[z ,ξ z ] 0 1 0 1 ∗ 1 1 lifts canonically to C -actions on P [r] and M (P [r],ν). We describe here the g,I,μ ∗ 1 C -ﬁxed loci of M (P [r],ν). g,I,μ ∗ 1 The C -ﬁxed loci of M (P [r],ν) are labeled by decorated graphs .The ver- g,I,μ tices v ∈ V() are decorated with a genus g(v) and the legs are labeled with markings in μ ∪ I ∪ ν.Eachvertex v is labeled by 1 1 0 ∈ P [r] or ∞∈ P [r]. Each edge e ∈ E() is decorated with a degree d that corresponds to the d -th power e e map 1 1 P [r]→ P [r]. The vertex labeling endows the graph with a bipartite structure. Avertexof over 0 ∈ P [r] corresponds to a stable map contracted to 0 given by an element of M (BZ ) with monodromies μ(v) speciﬁed by the corresponding g(v),I(v),μ(v) r entries of μ and the negatives of the degrees d of the incident edges (modulo r). We will use the notation r r M = M = M (BZ ). g(v),I(v),μ(v) r v g(v);I(v),μ(v) r r The fundamental class [M ] of M is of dimension v v dim M = 3g(v) − 3 + I(v) + μ(v) . As explained in Section 1.2.2, : M → M g(v),n(v) is a ﬁnite covering of the moduli space M where g(v),n(v) n(v) = μ(v) + I(v) is the total number of markings and edges incident to v. The stable map over ∞∈ P [r] can take two different forms. If the target does not expand, then the stable map has (ν) preimages of ∞∈ P [r]. Each preimage is described by an unstable vertex of labeled by ∞. If the target expands, then the stable map is a possibly disconnected rubber map. The ramiﬁcation data is given by incident edges over 0 of the rubber and by elements of ν over ∞ of the rubber. In this case every vertex v over ∞∈ P [r] describes a connected component of the rubber map. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 247 Let g(∞) be the genus of the possibly disconnected domain of the rubber map. We will denote the moduli space of stable maps to rubber by M and the virtual vir fundamental class by [M ] , vir dim M = 2g(∞) − 3 + n(∞), where n(∞) is the total number of markings and edges incident to vertices over ∞, n(∞) = I(∞) + (ν) +|E()|. vir The image of the virtual fundamental class [M ] in the moduli space of (not necessar- ily connected ) stable curves is denoted by DR . Avertex v ∈ V() is unstable when 2g(v) − 2 + n(v) ≤ 0. There are four types of unstable vertices: (i) v → 0, g(v) = 0, v carries no markings and one incident edge, (ii) v → 0, g(v) = 0, v carries no markings and two incident edges, (iii) v → 0, g(v) = 0, v carries one marking and one incident edge, (iv) v → ∞,g(v) = 0, v carries one marking and one incident edge. The target of the stable map expands iff there is at least one stable vertex over ∞. A stable map in the C -ﬁxed locus corresponding to is obtained by gluing to- gether maps associated to the vertices v ∈ V() with Galois covers associated to the 0 ∗ edges. Denote by V () the set of stable vertices of over 0. Then the C -ﬁxed locus st corresponding to is isomorphic to the product r ∼ 0 M × M , if the target expands, v∈V () v ∞ st M = 0 M , if the target does not expand, v∈V () v st quotiented by the automorphism group of and the product of cyclic groups Z associ- ated to the Galois covers of the edges. Thus the natural morphism corresponding to , ι : M → M P [r],ν , g,I,μ is of degree Aut() d , e∈E() onto the image ι(M ). Recall that D =|μ|=|ν| is thedegreeofthe stable map. We omit the data g(∞),I(∞), ν,and δ(∞) in the notation. By [3, Lemma 2.3], a double ramiﬁcation cycle vanishes as soon as there are at least two nontrivial components of the domain. However, we will not require the vanishing result here. Our analysis will naturally avoid disconnected domains mapping to the rubber over ∞∈ P [r]. 248 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Lemma 6. —For r > D, the unstable vertices of type (i) and (ii) can not occur. Proof. — At each stable vertex v ∈ V() over 0 ∈ P [r], the condition |μ |=0mod r has to hold in order for M (BZ ) to be non-empty. Because of the balancing con- g ,I ,μ r v v v dition at the nodes, the parts of μ at v have to add up to the sum of the degrees of the incident edges modulo r. Since r > D, the parts of μ at v must addupto exactly the sum of the degrees of the incident edges. So the incident edges of the stable vertices and the type (iii) unstable vertices must account for the entire degree D. Thus, there is no remaining edge degree for unstable vertices of type (i) and (ii). 2.4. Localization formula We write the C -equivariant Chow ring of a point as A (•) = Q[t], where t is the ﬁrst Chern class of the standard representation. For the localization formula, we will require the inverse of the C -equivariant Euler 1 ∗ class of the virtual normal bundle in M (P [r],ν) to the C -ﬁxed locus corresponding g,I,μ to .Let f : C → P [r], [f ]∈ M . The inverse Euler class is represented by 1 ∗ 1 e(H (C, f T 1 (−∞))) 1 1 P [r] (19) = . vir 0 ∗ e(H (C, f T 1 (−∞))) e(N ) e(N ) e(Norm ) P [r] i ∞ Formula (19) has several terms which require explanation. We assume r > |μ|=|ν|, so unstable vertices over 0 ∈ P [r] of type (i) and (ii) do not occur in . First consider the leading factor of (19), 1 ∗ e(H (C, f T (−∞))) P [r] (20) . 0 ∗ e(H (C, f T 1 (−∞))) P [r] To compute this factor one uses the normalization exact sequence for the domain C tensored with the line bundle f T 1 (−∞). The associated long exact sequence in co- P [r] homology decomposes the leading factor into a product of vertex, edge, and node con- tributions. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 249 • Let v ∈ V() be a stable vertex over 0 ∈ P [r] corresponding to a contracted component C . From the resulting stable map to the orbifold point C → BZ ⊂ P [r], v r we obtain an orbifold line bundle L = f T 1 | P [r] 0 on C which is an rth root of O . Therefore, the contribution v C 1 ∗ e(H (C , f T 1 | )) v P [r] 0 0 ∗ e(H (C , f T | )) v P [r] 0 yields the class ∗ (1/r) ∗ c −R π L ⊗ O ∈ A M ⊗ Q t, , rk ∗ g(v),I(v),μ(v) (1/r) where L → M is the universal rth root, O is a trivial line bundle g(v),I(v),μ(v) ∗ 1 with a C -action of weight ,and rk = g − 1 + E(v) is the virtual rank of −R π L. Unstable vertices of type (iii) over 0 ∈ P [r] and the vertices over ∞ contribute factors of 1. i ∗ • The edge contribution is trivial since the degree of f T 1 (−∞) is less than 1, P [r] see [24, Section 2.2]. • The contribution of a node N over 0 ∈ P [r] is trivial. The space of sec- 0 ∗ 1 tions H (N, f T 1 (−∞)) vanishes because N must be stacky, and H (N, P [r] f T 1 (−∞)) is trivial for dimension reasons. P [r] Nodes over ∞∈ P [r] contribute 1. Consider next the last two factors of (19), 1 1 e(N ) e(N ) i ∞ −1 1 • The product e(N ) is over all nodes over 0 ∈ P [r] of the domain C which are forced by the graph . These are nodes where the edges of are attached to the vertices. If N is a node over 0 forced by the edge e ∈ E() and the associated vertex v is stable, then t ψ (21) e(N) = − . rd r e 250 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE This expression corresponds to the smoothing of the node N of the domain curve: e(N) is the ﬁrst Chern classe of the normal bundle of the divisor of nodal domain curves. The ﬁrst Chern classes of the cotangent lines to the branches at the node are divided by r because of the orbifold twist, see Section 1.2. In the case of an unstable vertex of type (iii), the associated edge does not produce a node of the domain. The type (iii) edge incidences do not appear in −1 e(N ) . • N corresponds to the target degeneration over ∞∈ P [r].The factor e(N ) is ∞ ∞ 1ifthe target (P [r], ∞) does not degenerate and t + ψ e(N ) =− e∈E() if the target does degenerate [17]. Here, ψ is the ﬁrst Chern class of the cotan- gent line bundle to the bubble in the target curve at the attachment point to ∞ of P [r]. Summing up our analysis, we write out the outcome of the virtual localization formula [15]in A (M (P [r],ν)) ⊗ Q[t, ].Wehave ∗ g,I,μ vir 1 1 [M ] vir (22) M P [r],ν = · ι , g,I,μ ∗ vir |Aut()| d e(Norm ) e∈E() vir [M ] where is the product of the following factors: vir e(Norm ) r t ∗ g(v)−1+|E(v)|−d • · c (−R π L)( ) for each stable vertex v ∈ V() d ∗ e∈E(v) d≥0 −ψ r over 0, e∈E() •− if the target degenerates. t+ψ Remark 2. —Aswementioned in Remark 1, there are two conventions for the orbifold structure on the space of rth roots. In the paper, we follow the convention which assigns to each node of the curve the stabilizer Z/rZ. It is also possible to assign to a node of type (a , a ) the stabilizer Z/qZ,where q is the order of a and a in Z/rZ.Welisthere the required modiﬁcations of our formulas when the latter convention is followed. Denote p = gcd(r, a ) = gcd(r, a ).Wethenhave pq = r.InSection 1.2.4,the ﬁrst Chern class of the normal line bundle to a boundary divisor in the space of rth roots becomes −(ψ + ψ )/q rather than −(ψ + ψ )/r. The degrees of the maps φ and ϕ in Equations (14)and (15)are equalto p instead of 1. The factor in the last term of |E( )| Chiodo’s formula (16) becomes . Because of the above, the factor r in Proposition 4 |E( )| is transformed into q . The statement of Corollary 4 is unchanged, but in the proof |E( )| |E( )| |E( )| the factor r is now obtained as a product of q from Proposition 4 and p from the degree of φ and ϕ. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 251 In the localization formula, the Euler classes of the normal bundles at the nodes ψ ψ t e t e over 0 become − instead of − . On the other hand, at each node there are p qd q rd r e e ways to reconstruct an rth root bundle from its restrictions to the two branches. Thus the contribution of each node still has an r in the numerator. e(N) 2.5. Extracting the double ramiﬁcation cycle 2.5.1. Three operations We will now perform three operations on the localization formula (22)for the 1 vir virtual class [M (P [r],ν)] : g,I,μ (i) the C -equivariant push-forward via (23) : M P [r],ν → M g,I,μ g,n to the moduli space M with trivial C -action, g,n −1 (ii) extraction of the coefﬁcient of t after push-forward by , (iii) extraction of the coefﬁcient of r . −1 After push-forward by , the coefﬁcient of t is equal to 0 because vir 1 ∗ M P [r],ν ∈ A (M ) ⊗ Q[t]. ∗ g,I,μ g,n −1 Using the result of Section 1.4,all termsofthe t coefﬁcient will be seen to be polynomi- als in r, so operation (iii) will be well-deﬁned. After operations (i)–(iii), only two nonzero terms will remain. The cancellation of the two remaining terms will prove Theorem 1. To perform (i)–(iii), we multiply the -push-forward of the localization formula (22) 0 0 by t and extract the coefﬁcient of t r . To simplify the computations, we introduce the new variable s = tr. 0 0 0 0 Then, instead of extracting the coefﬁcient of t r ,weextract thecoefﬁcientof s r . 2.5.2. Push-forward to M g,n For each vertex v ∈ V(),wehave r r M = M . v g(v);I(v),μ(v) Following Section 1.2.2, we denote the map to moduli by (24) : M → M . g(v),n(v) g(v);I(v),μ(v) 252 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE The use of the notation in (23)and (24) is compatible. Denote by 2d−2g(v)+1 ∗ d (25) c = r c −R π L ∈ R (M ). d ∗ d ∗ g(v),n(v) By Proposition 5, c is a polynomial in r for r sufﬁciently large. From (22) and the contribution calculus for presented in Section 2.5.2,wehave a complete formula for the C -equivariant push-forward of t times the virtual class: vir (26) t M P [r],ν ∗ g,I,μ vir s 1 1 [M ] = · · ι , ∗ ∗ vir r |Aut()| d e(Norm ) e∈E() vir [M ] where ι is the product of the following factors: ∗ ∗ vir e(Norm ) • afactor r d 1 g(v)−d d · · c s ∈ R (M ) ⊗ Q s, d g(v),n(v) rd s 1 − ψ s e∈E(v) s d≥0 for each stable vertex v ∈ V() over 0, • afactor e∈E() − · · DR s 1 + ψ if the target degenerates. 2.5.3. Extracting coefﬁcients Extracting the coefﬁcient of r .— By Proposition 5, the classes c are polynomial in r for r sufﬁciently large. We have an r in the denominator in the prefactor on the right side of (26) which comes from the multiplication by t on the left side. However, in all other factors, we only have positive powers of r, with at least one r per stable vertex of the graph over 0 and one more r if the target degenerates. The only graphs which contribute to the coefﬁcient of r are those with exactly one r in the numerator. There are only two graphs which have exactly one r factor in the numerator: • the graph with a stable vertex of full genus g over 0 and (ν) type (iv) unstable vertices over ∞, • the graph with a stable vertex of full genus g over ∞ and (μ) type (iii) unstable vertices over 0. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 253 No terms involving ψ classes contribute to the r coefﬁcient of either or since every ψ class in the localization formula comes with an extra factor of r.Wecan now write the r coefﬁcient of the right side of (26): vir 1 g−d (27)Coeff 0 t M P [r],ν = Coeff 0 c s − DR (A) r ∗ g,I,μ r d g d≥0 d 1 in R (M ) ⊗ Q[s, ]. g,n Extracting the coefﬁcient of s .— The remaining powers of s in (27) appear only with the classes c in the contribution of the graph . In order to obtain s ,wetaketotake d = g, vir 0 0 0 (28)Coeff t M P [r],ν = Coeff [ c ]− DR (A) ∗ g,I,μ g g s r r in R (M ). g,n 2.5.4. Final relation 1 vir 0 0 Since Coeff [ (t[M (P [r],ν)] )] vanishes, we can rewrite (28)as ∗ g,I,μ s r 2g−2g+1 ∗ g (29)DR (A) = Coeff r c −R π L ∈ R (M ), g r ∗ g ∗ g,n using deﬁnition (25)of c .ByProposition 5 applied to the right side of relation (29), −g g g DR (A) = 2 P (A) ∈ R (M ). g g,n The proof of Theorem 1 is complete. 3. A few applications 3.1. A new expression for λ We discuss here a few more examples of the formula for Corollary 3: a formula for the class λ supported on the boundary divisor with a nonseparating node. We give the formulas for the class λ in M up to genus 4. g g The answer is expressed as a sum of labeled stable graphs with coefﬁcients where each stable graph has at least one nontrivial cycle. The genus of each vertex is written inside the vertex. The powers of ψ -classes are written on the corresponding half-edges (zero powers are omitted). The expressions do not involve κ -classes. Each labeled graph describes a moduli space M (a product of moduli spaces associated with its vertices), a tautological class α ∈ R (M ) and a natural map π : M → M . Our convention is that then represents the cycle class π α. For instance, if g ∗ the graph carries no ψ -classes, the class α equals 1 and the map π is of degree |Aut( )|. The cycle class represented by is then |Aut( )| times the class of the image of π . 254 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Genus 1. λ = . Genus 2. 1 1 λ = + . 2 1 0 240 1152 Genus 3. 1 1 1 1 ψ ψ λ = + − + 3 2 2 1 1 1 2016 2016 672 5760 13 1 1 − − + . 0 1 0 1 0 30240 5760 82944 Genus 4. 1 1 1 1 3 2 ψ ψ 2 λ = + − − 4 3 3 1 2 1 2 11520 3840 2880 3840 1 1 1 1 ψ ψ ψ ψ − − − − 1 2 1 2 1 2 1 2 ψ ψ 1440 1920 2880 3840 1 1 1 1 ψ ψ ψ + ψ + + + 2 2 2 1 ψ 1 48384 48384 115200 960 23 1 1 − − − 2 0 2 0 2 0 ψ ψ 100800 57600 16128 1 1 1 − − − 2 0 1 1 1 1 ψ ψ 16128 57600 16128 1 23 23 − − + 1 1 1 1 0 2 ψ ψ 16128 100800 100800 0 1 23 1 1 + + + 1 1 1 0 1 0 1 50400 16128 115200 1 13 1 + − − 1 1 0 1 0 276480 725760 138240 43 13 1 − − − 1 0 1 0 1 0 1612800 725760 276480 7962624 DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 255 All these expressions are obtained by taking n = 0 in the formula for the double ramiﬁcation cycle. 3.2. Hodge integrals Corollary 3 may be applied to any Hodge integral which contains the top Chern class of the Hodge bundle. As an example, we consider the following Hodge integral (ﬁrst calculated in [10]) related to the constant map contribution in the Gromov-Witten theory of Calabi-Yau threefolds. Proposition 7. —For g ≥ 1, we have 1 1 B B 2g 2g+2 λ λ λ =− . g+1 g g−1 2 (2g)! 2g 2g + 2 g+1 Proof. — We compute the integral by replacing λ with the expression obtained g+1 from Corollary 3. g+1 In Pixton’s formula for DR (∅) = (−1) λ , we only have to consider graphs g+1 g+1 with no separating edges. On the other hand, λ λ vanishes on strata as soon as there g g−1 are at least two nonseparating edges. Thus we are left with just one graph: the graph with a vertex of genus g and one loop. g+1 The edge term in Pixton’s formula for (−1) DR (∅) = λ for the unique g+1 g+1 remaining graph is (ψ +ψ ) 2k+2 1 − e a g+1 g (−1) = (−1) ψ + ψ . k+1 ψ + ψ 2 (k + 1)! k≥0 The contribution of the graph is therefore the r-free term of r−1 g 2g+2 (−1) a · (ψ + ψ ) λ λ . 1 2 g g−1 g+1 2r 2 (g + 1)! g,2 a=0 1 1 The factor of comes from the automorphism group of the graph, and the factor of 2 r comes from the ﬁrst Betti number of graph. Since r−1 2g+2 2 a = B r + O r , 2g+2 a=0 we see that the r-free term equals 2g+2 g g (−1) (ψ + ψ ) λ λ . 1 2 g g−1 g+2 2 (g + 1)! g,2 256 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Using Lemma 8 below we can rewrite this expression as B B 1 1 1 B B 2g+2 2g 2g 2g+2 g g+1 (−1) · (−1) =− , g+2 2 (g + 1)! 2g (2g − 1)!! 2 (2g)! 2g 2g + 2 which completes the derivation. Lemma 8. —For g ≥ 1, we have B 1 2g g g+1 (ψ + ψ ) λ λ = (−1) . 1 2 g g−1 2g (2g − 1)!! g,2 Proof. — According to Faber’s socle formula [9, 14], we have g+1 (−1) B p q 2g (30) ψ ψ λ λ = g g−1 1 2 2g 2 g(2p − 1)!!(2q − 1)!! g,2 whenever p + q = g. The formula holds even if p or q is equal to 0. After expanding (ψ + ψ ) and applying (30), we obtain 1 2 g! 1 2g g g+1 (ψ + ψ ) λ λ = (−1) . 1 2 g g−1 2g g 2 p!(2p − 1)!!q!(2q − 1)!! g,2 p+q=g The right side is, up to a factor, the sum of even binomial coefﬁcients in the 2g-th line of the Pascal triangle: g g 1 2 2 2g = = p!(2p − 1)!!q!(2q − 1)!! (2p)!(2q)! (2g)! 2p p+q=g p+q=g p=0 3g−1 = . (2g)! Finally, we have 3g−1 B g! 2 B 1 2g 2g g+1 g+1 (−1) · = (−1) 2g g 2 (2g)! 2g (2g − 1)!! as claimed. The socle formula is true if at most one power of a ψ class vanishes while the powers of all other ψ classes are positive. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 257 3.3. Local theory of curves The local Gromov-Witten theory of curves as developed in [2] starts with a fun- damental pairing with a double ramiﬁcation cycle (the resulting matrix then also appears in the quantum cohomology of the Hilbert scheme of points of C [30]). We rederive the double ramiﬁcation cycle pairing as an almost immediate consequence of Theorem 1. Proposition 9. —For g ≥ 1, we have 2g a B 2g g+1 λ λ = (−1) . g g−1 (2g)! 2g DR (a,−a) Proposition 9 is essentially equivalent to the basic calculation of [2, Theorem 6.5] in the Gromov-Witten theory of local curves. There are two minor differences. First, our answer differs from [2, Theorem 6.5] by a factor of 2g coming from the choice of branch point in [2, Theorem 6.5]. Second, if we write out the generating series obtained from Proposition 9 then we only obtain the cot(u) summand in the formula of [2,Theo- rem 6.5] after a standard correction accounting for the geometric difference between the space of rubber maps and the space of maps relative to three points. Proof. — We need only compute the contributions to DR (a, −a) of all rational tailstypegraphssince λ λ vanishes on all other strata. Since we just have two marked g g−1 points, there are exactly two rational tails type graphs: • the graph with a single vertex, • the graph with a vertex of genus g, a vertex of genus 0 containing both marked points, and exactly one edge joining the two vertices. The contribution of the ﬁrst graph is (using Lemma 8) 2g 2g a a B 1 2g g g+1 (ψ + ψ ) λ λ = · (−1) 1 2 g g−1 g g 2 g! 2 g! 2g (2g − 1)!! g,2 2g a B 2g g+1 = (−1) . (2g)! 2g The contribution of the second graph vanishes because the weight of the edge is equal to 0. For the above computation, we only require Hain’s formula for the restriction of the ct double ramiﬁcation cycle to M (the extension to M is not needed). In the forthcom- g,n g,n ing paper [22], we will study the Gromov-Witten theory of the rubber over the resolution of the A -singularity where the full structure is needed. n 258 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Acknowledgements We are grateful to A. Buryak, R. Cavalieri, A. Chiodo, E. Clader, C. Faber, G. Farkas, S. Grushevsky, J. Guéré, G. Oberdieck, D. Oprea, R. Vakil, and Q. Yin for discussions about double ramiﬁcation cycles and related topics. The Workshop on Pixton’s conjectures at ETH Zürich in October 2014 played an important role in our collaboration. A. P. and D. Z. have been frequent guests of the Forschungsinstitut für Mathematik (FIM) at ETHZ. The paper was completed at the conference Moduli spaces of holomorphic differen- tials at Humboldt University in Berlin in February 2016 attended by all four authors. F. J. was partially supported by the Swiss National Science Foundation grant SNF-200021143274. R. P. was partially supported by SNF-200021143274, SNF- 200020162928, ERC-2012-AdG-320368-MCSK, SwissMap, and the Einstein Stiftung. A. P. was supported by a fellowship from the Clay Mathematics Institute. D. Z. was supported by the grant ANR-09-JCJC-0104-01. Appendix A: Polynomiality by A. Pixton A.1 Overview Our goal here is to present a self contained proof of Proposition 3 of Section 1.4.Let A = (a ,..., a ) 1 n be a vector of k-twisted double ramiﬁcation data for genus g ≥ 0. A longer treatment, in d,k the context of the deeper polynomiality of P (A) in the parts a of A, will be given in [35]. Proposition 3 (Pixton [35]). — Let ∈ G be a ﬁxed stable graph. Let Q be a polynomial g,n with Q-coefﬁcients in variables corresponding to the set of half-edges H( ). Then the sum F(r) = Q w(h ), ...,w(h ) 1 |H( )| w∈W ,r,k h ( ) over all possible k-weightings mod r is (for all sufﬁciently large r) a polynomial in r divisible by r . A.2 Polynomiality of F(r) A.2.1 Ehrhart polynomials The polynomiality of F(r) for r sufﬁciently large will be derived as a consequence of a small variation of the standard theory [37] of Ehrhart polynomials of integral convex polytopes. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 259 Let M be a v ×hmatrix with Q-coefﬁcients satisfying the following two properties: (i) M is totally unimodular: every square minor of M has determinant 0, 1, or −1. (ii) If x = (x ) is a column vector satisfying j 1≤j≤h Mx =0and x ≥0forall j, then x = 0. We will use the notation x ≥ 0 to signify x ≥ 0for all j.Let h h Z = x | x ∈ Z and x ≥ 0 . ≥0 Proposition A1. —Let a, b ∈ Z be column vectors with integral coefﬁcients. Let Q ∈ Q[x ,..., x ] be a polynomial. Then, the sum over lattice points 1 h S(r) = Q(x) x∈Z , Mx=a+rb ≥0 is a polynomial in the integer parameter r (for all sufﬁciently large r). Proof. — We follow the treatment of Ehrhart theory in [37, Sections 4.5–4.6]. To start, we may assume all coefﬁcients a of a are nonnegative: if not, multiply the negative coefﬁcients of a and the corresponding parts of M and b by −1. Next, we deﬁne a generating function of h + v + 1 variables: R(X ,..., X , Y ,..., Y , Z) 1 h 1 v x y z = X Y Z . h v x∈Z , y∈Z , z∈Z , Mx=y+zb ≥0 ≥0 ≥0 By [37, Theorem 4.5.11], R represents a rational function of the variables X, Y, and Z with denominator given by a product of factors of the form 1 − Monomial(X, Y, Z). The monomials in the denominator factors correspond to the extremal rays of the cone h+v+1 in Z speciﬁed by the condition ≥0 Mx = y + zb. Because M is totally unimodular, these extremal rays all have integral generators with z-coordinate equal to 0 or 1. Therefore, the denominator is a product of factors of the form 1 − Monomial(X, Y) and 1 − Monomial(X, Y) · Z. The standard situation for the Ehrhart polynomial occurs when Q = 1 is the constant polynomial and a = 0. The variation required here is not very signiﬁcant, but we were not able to ﬁnd an adequate reference. 260 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE The sum S(r) in the statement of Proposition A1 can be obtained from R by exe- cuting the following steps: (i) apply a differential operator in the X variables to R which has the effect of x y z multiplying the coefﬁcient of X Y Z by Q(x), (ii) apply the differential operator a a a 1 2 v 1 ∂ ∂ ∂ ... , v a a 1 2 a ! ∂ Y ∂ Y ∂ Y 1 2 i=1 v (iii) set X = 1for all1 ≤ j ≤ hand Y = 0for all1 ≤ i ≤ v, j i (iv) extract the coefﬁcient of Z . After step (iii) and before step (iv), we have a rational function in Z with denominator equal to a power of 1 − Z. The Z coefﬁcient of such a rational function is (eventually) a polynomial in r. A.2.2 Vertex-edge matrix We now prove the polynomiality of F(r) for all r sufﬁciently large. Let A = (a ,..., a ) 1 n be a vector of k-twisted double ramiﬁcation data for genus g ≥ 0. Let ∈ G be a ﬁxed g,n stable graph. For all sufﬁciently large r, the sum over half-edge weightings (31) w ∈ W , 0 ≤ w(h)< r ,r,k satisfying conditions mod r, (32)F(r) = Q w(h ), ...,w(h ) , 1 |H( )| w∈W ,r,k can be split into ﬁnitely many sums over half-edge weightings satisfying equalities (in- volving multiples of r). We study each sum in the splitting of (32) separately. The conditions on the weights w(h) in these sums will be of the following form. The inequalities 0 ≤ w(h)< r continue to hold. Moreover, each leg, edge, and vertex yields an equality: By property (ii) of the deﬁnition of M, the specialization is well deﬁned. There are no denominator factors in R of the form 1 − Monomial(X). Equalities not equalities mod r . DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 261 (i) w(h ) = a + rb at the ith leg h ,where i i i i b =0if a ≥0and b =1if a < 0, i i i i (ii) w(h) + w(h ) = 0or w(h) + w(h ) = r along the edge (h, h ), (iii) w(h) = k(2g(v) − 2 + n(v)) + rb at the vertex v with b ∈ Z. v v v(h)=v The conditions w(h )< r are now redundant with the equalities of type (i) and can be removed. Along an edge (h, h ), the condition w(h)< r only serves to rule out the possibility w(h) = r, w(h ) = 0 (if we have the equality w(h) + w(h ) = r). Thus the condition w(h)< r can be removed if we subtract off the contributions with w(h) = r, w(h ) = 0. These contributions are given by more sums of the same form (corresponding to the graph formed by deleting the edge (h, h ) from ), multiplied by powers of r and 0 coming from factors of w(h) and w(h ) in the polynomial Q. Thus the original sum F(r) can be split into a linear combination (with coefﬁcients being polynomials in r) of sums with weight conditions of the following form: inequalities w(h) ≥ 0 and equalities of the form (i), (ii), and (iii) associated to legs, edges, and vertices as de- scribed above. We must check that the resulting sums satisfy the hypotheses of Proposition A1. Consider the graph obtained from by adding a new vertex in the middle of each edge e ∈ E( ) and at the end of each leg l ∈ L( ). The half-edge set of then corresponds bijectively to the edge set of , H( ) ←→ E . We associate each linear condition on the half-edge weights w(h) described in (i)–(iii) above to the corresponding vertex of . For types (i) and (ii), the corresponding vertex of is a new vertex. For (iii), the corresponding vertex already exists in . The matrix M associated to the linear terms in these linear conditions is precisely the vertex-edge incidence matrix of :Mis av × hmatrix, v = V , h = E = H( ) , with matrix entry 1 if the corresponding vertex of meets the corresponding edge of (and 0 otherwise). Since all the entries of M are nonnegative and every column contains a positive entry, we immediately see (Mx = 0and x ≥ 0) ⇒ x = 0. Because is bipartite, the incidence matrix M is totally unimodular by Proposition A2 below. We may therefore apply Proposition A1 to each sum of the splitting of (32). Each such sum is a polynomial in r for sufﬁciently large r.Hence, F(r) is a polynomial in r for sufﬁciently large r. 262 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Proposition A2. — The vertex-edge matrix of a bipartite graph is totally unimodular. A proof of general results concerning the total unimodularity of vertex-edge ma- trices can be found in [20]. Proposition A2 arises as a special case. h ( ) A.3 Divisibility by r In this section we write b = h ( ) for convenience. We wish to show that the polynomial F(r) is divisible by r . This is implied by checking that the valuation condition, val F(p) ≥ b, holds for all sufﬁciently large primes p.Here, val : Q → Z is the p-adic valuation. Let h =|H( )| be the number of half-edges of . We write the evaluation F(p) as F(p) = Q(w ,...,w ) 1 h 0≤w ,...,w ≤p−1 1 h where the sum is over h-tuples (w ,...,w ) subject to h − b linear conditions mod p. 1 h The polynomial Q has (p-integral) Q-coefﬁcients. We count the linear conditions mod p as follows. By the deﬁnition of a k-weighting mod p, we start with v = L( ) + E( ) + V( ) linear conditions, but the conditions are not independent since the vector A of k-twisted double ramiﬁcation data satisﬁes a = k(2g − 2 + n). i=1 After removing a redundant linear relation, we have v − 1 independent linear conditions. Since h = H( ) = L( ) + 2 E( ) , and is connected, we see h − b = h − E( ) − V( ) + 1 = v − 1. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 263 Let Q be a polynomial with Q-coefﬁcients in h variables, with all coefﬁcients p-integral. We will prove every summation (33) F(p) = Q(w ,...,w ), 1 h 0≤w ,...,w ≤p−1 1 h where the sum is over h-tuples (w ,...,w ) subject to h − b linear conditions mod p,is 1 h divisible by p for p ≥ deg(Q) + 2. We maycertainlytake Q to be a monomial. Without loss of generality, we need only consider Q(w ,...,w ) = w . 1 h j j=1 We write the linear conditions imposed in the sum (33)as m w + =0mod p ij j i j=1 for 1 ≤ i ≤ h − b. Alternatively, we may impose the conditions by standard p-th root of unity ﬁlters: m w h−b ij j p F(p) = w · z · z . i i z ,...,z 0≤w ,...,w ≤p−1 j=1 1≤i≤h−b 1≤i≤h−b 1 h−b 1 h 1≤j≤h The ﬁrst sum on the right is over all (h − b)-tuples (z ,..., z ) of p-th roots of unity. 1 h−b The second sum on the right is over all h-tuples (w ,...,w ). 1 h Consider the following polynomial with integer coefﬁcients: p p+1 Z − pZ + (p − 1)Z G(Z) = aZ = . (1 − Z) 0≤a≤p−1 We can use G to rewrite the terms inside the ﬁrst sum in the above expression: ij h−b (34) p F(p) = G z · z . i i z ,...,z 1≤j≤h 1≤i≤h−b 1≤i≤h−b 1 h−b 24 2 To increase w to w , add a new variable w and a new linear condition w = w . If a variable w does not 1 h+1 1 h+1 j appear in Q, then a reduction to a fewer variable case can be made. 264 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE p−2 For every p-th root of unity z,G(z) has p-adic valuation at least : either z = 1 p−1 p(p−1) and G(z) = or p 1 z = 1 ⇒ G(z) = and val (z − 1) = . z − 1 p − 1 Therefore, every term in the sum on the right side of (34)has p-adic valuation at least p−2 h · .Hence, p−1 p − 2 val F(p) ≥ h · − (h − b). p − 1 Since val (F(p)) is an integer, val F(p) ≥ b if p ≥ h + 2. d,k A.4 Polynomiality of P (A) in the a For double ramiﬁcation data of ﬁxed length n,consider d,k d P (a ,..., a ) ∈ R (M ) 1 n g,n as a function of (a ,..., a ) ∈ Z , a = k(2g − 2 + n) 1 n i i=1 holding g, d,and k ﬁxed. The following property is proven in [35]: d,k P (a ,..., a ) is polynomial in the variables a . 1 n i By Theorem 1, the double ramiﬁcation cycle DR (A) is then also polynomial in the variables a . Shadows of the polynomiality of the double ramiﬁcation cycle were proven earlier in [3, 29]. We view the p-th root of unity z as lying in Q .The p-adic valuation of x = z − 1 can be calculated from the minimal polynomial p−1 (x + 1) = 0. i=0 DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 265 REFERENCES 1. D. ABRAMOVICH,T.GRABER and A. VISTOLI, Gromov-Witten theory of Deligne-Mumford stacks, Am. J. Math., 130 (2008), 1337–1398. 2. J. BRY AN and R. PANDHARIPANDE, The local Gromov-Witten theory of curves, J. Am. Math. Soc., 21 (2008), 101–136. 3. A. BURYAK,S.SHADRIN,L.SPITZ and D. ZVONKINE,Integrals of ψ -classes over double ramiﬁcation cycles, Am. J. Math., 137 (2015), 699–737. rt 4. R. CAVALIERI,S.MARCUS and J. WISE, Polynomial families of tautological classes on M , J. Pure Appl. Algebra, 216 g,n (2012), 950–981. 5. A. CHIODO, Stable twisted curves and their r -spin structures, Ann. Inst. Fourier, 58 (2008), 1635–1689. 6. A. CHIODO, Towards an enumerative geometry of the moduli space of twisted curves and rth roots, Compos. Math., 144 (2008), 1461–1496. 7. A. 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MUMFORD, Towards an enumerative geometry of the moduli space of curves, in M. Artin and J. Tate (eds.) Arithmetic and Geometry, Part II, pp. 271–328, Birkhäuser, Basel, 1983. 29. A. OKOUNKOV and R. PANDHARIPANDE, Gromov-Witten theory, Hurwitz numbers, and completed cycles, Ann. Math., 163 (2006), 517–560. 30. A. OKOUNKOV and R. PANDHARIPANDE, Quantum cohomology of the Hilbert scheme of points of the plane, Invent. Math., 179 (2010), 523–557. 31. R. PANDHARIPANDE and A. PIXTON, Relations in the tautological ring of the moduli space of curves, arXiv:1301.4561. 32. R. PANDHARIPANDE,A.PIXTON and D. ZVONKINE,Relations on M via 3-spin structures, J. Am. Math. Soc., 28 (2015), g,n 279–309. 266 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE 33. A. PIXTON, Conjectural relations in the tautological ring of M , arXiv:1207.1918. g,n 34. A. PIXTON, Double ramiﬁcation cycles and tautological relations on M ,available from theauthor. g,n 35. A. PIXTON, On combinatorial properties of the explicit expression for double ramiﬁcation cycles, in preparation. 36. B. RIEMANN, Theorie der Abel’schen Functionen, J. Reine Angew. Math., 54 (1857), 115–155. 37. R. STANLEY, Enumerative Combinatorics: Vol I, Cambridge University Press, Cambridge, 1999. 38. G. van der GEER, Cycles on the moduli space of abelian varieties, in Moduli of Curves and Abelian Varieties, Aspects Math., vol. E33, pp. 65–89, Vieweg, Braunschweig, 1999. F. J. Department of Mathematics, University of Michigan, Ann Arbor, USA janda@umich.edu R. P. Departement Mathematik, ETH Zürich, Zürich, Switzerland rahul@math.ethz.ch A. P. Department of Mathematics, MIT, Cambridge, USA apixton@mit.edu D. Z. Institut de Mathématiques de Jussieu, CNRS, Paris, France dimitri.zvonkine@imj-prg.fr Manuscrit reçu le 27 mars 2016 Manuscrit accepté le 20 mars 2017 publié en ligne le 10 mai 2017. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals http://www.deepdyve.com/lp/springer-journals/double-ramification-cycles-on-the-moduli-spaces-of-curves-Sk4oluAFq0

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Publisher: Springer Journals
Copyright: Copyright © 2017 by IHES and Springer-Verlag Berlin Heidelberg
Subject: Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN: 0073-8301
eISSN: 1618-1913
DOI: 10.1007/s10240-017-0088-x
Publisher site: See Article on Publisher Site

Abstract

DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES by F. JANDA, R. PANDHARIPANDE, A. PIXTON, and D. ZVONKINE ABSTRACT 1 1 1 Curves of genus g which admit a map to P with speciﬁed ramiﬁcation proﬁle μ over 0 ∈ P and ν over ∞∈ P deﬁne a double ramiﬁcation cycle DR (μ, ν) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramiﬁcation cycles on the moduli space of Deligne-Mumford stable curves. The cycle DR (μ, ν) for stable curves is deﬁned via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for DR (μ, ν) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramiﬁcation cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case. When μ = ν =∅, the formula for double ramiﬁcation cycles expresses the top Chern class λ of the Hodge bundle of M as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given. 0. Introduction 0.1. Tautological rings Let M be the moduli space of stable curves of genus g with n marked points. g,n SinceMumford’s article[28], there has been substantial progress in the study of the structure of the tautological rings ∗ ∗ R (M ) ⊂ A (M ) g,n g,n of the moduli spaces of curves. We refer the reader to [12]for asurvey of thebasic deﬁnitions and properties of R (M ). g,n The recent results [21, 31, 32] concerning relations in R (M ), conjectured to g,n be all relations [33], may be viewed as parallel to the presentation A Gr(r, n) = Q c (S), ..., c (S) / s (S), ..., s (S) 1 r n−r+1 n of the Chow ring of the Grassmannian via the Chern and Segre classes of the universal subbundle S → Gr(r, n). The Schubert calculus for the Grassmannian contains several explicit formulas for geo- metric loci. Our main result here is an explicit formula for the double ramiﬁcation cycle in R (M ). g,n All Chow groups will be taken with Q-coefﬁcients. DOI 10.1007/s10240-017-0088-x 222 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE 0.2. Double ramiﬁcation cycles 0.2.1. Notation Double ramiﬁcation data for maps will be speciﬁed by a vector A = (a ,..., a ), a ∈ Z 1 n i satisfying the balancing condition a = 0. i=1 The integers a are the parts of A. We separate the positive, negative, and 0 parts of A as follows. The positive parts of A deﬁne a partition μ = μ + ··· + μ . 1 (μ) The negatives of the negative parts of A deﬁne a second partition ν = ν + ··· + ν . 1 (ν) Up to a reordering of the parts, we have (1)A = (μ ,...,μ , −ν ,..., −ν , 0,..., 0 ). 1 (μ) 1 (ν) n−(μ)−(ν) Since the parts of A sum to 0, the partitions μ and ν must be of the same size. Let D =|μ|=|ν| be the degree of A. Let I be the set of markings corresponding to the 0 parts of A. The various constituents of A are permitted to be empty. The degree 0 case occurs when μ = ν =∅. If I =∅,then n = (μ) + (ν). The empty vector A is permitted, then n = 0, D = 0, and μ = ν = I =∅. There will be no disadvantage in assuming A is ordered as in (1). DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 223 0.2.2. Nonsingular curves Let M ⊂ M be the moduli space of nonsingular pointed curves. Let g,n g,n A = (a ,..., a ) 1 n be a vector of double ramiﬁcation data as deﬁned in Section 0.2.1.Let Z (A) ⊂ M g g,n be the locus parameterizing curves [C, p ,..., p ]∈ M satisfying 1 n g,n (2) O a p = O . C i i C Condition (2) is algebraic and deﬁnes Z (A) canonically as a substack of M . g g,n If [C, p ,..., p ]∈ Z (A), the deﬁning condition (2) yields a rational function (up 1 n g to C -scaling), (3) f : C → P , 1 1 of degree D with ramiﬁcation proﬁle μ over 0 ∈ P and ν over ∞∈ P . The markings corresponding to 0 parts lie over P \{0, ∞}. Conversely, every such morphism (3), up to C -scaling, determines an element of Z (A). We may therefore view Z (A) ⊂ M as the moduli space of degree D maps (up g g,n to C -scaling), f : C → P , with ramiﬁcation proﬁles μ and ν over 0 and ∞ respectively. The term double ramiﬁcation is motivated by the geometry of the map f . The dimension of Z (A) is easily calculated via the theory of Hurwitz covers. Ev- ery map (3) can be deformed within Z (A) to a map with only simple ramiﬁcation over 1 1 P \{0, ∞}. The number of simple branch points in P \{0, ∞} is determined by the Riemann-Hurwitz formula to equal (μ) + (ν) + 2g − 2. The dimension of the irreducible components of Z (A) can be calculated by varying the branch points [36] to equal (μ) + (ν) + 2g − 2 + (I) − 1 = 2g − 3 + n. The −1 on the left is the effect of the C -scaling. Hence, Z (A) is of pure codimension g in M . g,n When considering Z (A),we always assume the stability condition 2g − 2 + n > 0holds. g 224 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE 0.2.3. The Abel-Jacobi map ct Let M ⊂ M be the moduli space of curves of compact type. The universal g,n g,n Jacobian ct Jac → M g,n g,n is the moduli space of line bundles on the universal curve of degree 0 on every irreducible component of every ﬁber. There is a natural compactiﬁcation of Z (A) in the moduli space of compact type curves, ct ct Z (A) ⊂ Z (A) ⊂ M , g g,n obtained from the geometry of the universal Jacobian. Consider the universal curve over the compact type moduli space ct ct ct π : C → M . g,n ct The double ramiﬁcation vector A = (a ,..., a ) determines a line bundle L on C of 1 n relative degree 0, L = a [p ], i i ct ct where p ⊂ C is the section of π corresponding to the marking p . i i ct By twisting L by components of the π inverse images of the boundary divisors of ct M , we can easily construct a line bundle L which has degree 0 on every component g,n ct ct of every ﬁber of π . The result L is unique up to twisting by the π inverse images of ct the boundary divisors of M . g,n Via L , we obtain a section of the universal Jacobian, ct (4) φ : M → Jac . g,n g,n ct Certainly the closure of Z (A) in M lies in the φ inverse image of the 0-section S ⊂ g,n Jac of the relative Jacobian. We deﬁne g,n ct −1 ct (5) Z (A) = φ (S) ⊂ M . g g,n ct The substack Z (A) is independent of the choice of L . From the deﬁnitions, we see ct Z (A) ∩ M = Z (A). g,n g g DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 225 ct While S ⊂ Jac is of pure codimension g, Z (A) typically has components of g,n excess dimension (obtained from genus 0 components of the domain). Acycle classof the expected dimension vir ct ct g ct DR (A) = Z (A) ∈ A M g g g,n ∗ ct is deﬁned by φ ([S]). A closed formula for DR (A) was obtained by Hain [19]. A simpler approach was later provided by Grushevsky and Zakharov [18]. 0.2.4. Stable maps to rubber Let A = (a ,..., a ) be a vector of double ramiﬁcation data. The moduli space 1 n M P ,μ,ν g,I parameterizes stable relative maps of connected curves of genus g to rubber with ramiﬁcation proﬁles μ, ν over the point 0, ∞∈ P respectively. The tilde indicates a rubber target. We refer the reader to [25, 26] for the foundations. There is a natural morphism : M P ,μ,ν → M g,I g,n forgetting everything except the marked domain curve. Let R/{0, ∞} be a rubber target, and let f : C → R/{0, ∞} ∈ M P ,μ,ν g,I be a moduli point. If C is of compact type and R = P ,then ct [ f ] ∈ Z (A) ⊂ M . g,n Hence, we have the inclusion ct Z (A) ⊂ Im() ⊂ M . g,n 1 ∼ The virtual dimension of M (P ,μ,ν) is 2g − 3 + n.Wedeﬁne the double rami- g,I ﬁcation cycle to be the push-forward ∼ vir 1 g DR (A) = M P ,μ,ν ∈ A (M ). g ∗ g,I g,n 4 ct Hence, Z (A) ⊂ Z (A) is not always dense. 5 1 R is a chain of P s. 226 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE 0.2.5. Basic properties The ﬁrst property of DR (A) is a compatibility with the Abel-Jacobi construction of Section 0.2.3.Let ct ι : M → M g,n g,n be the open inclusion map. Proposition 1 (Marcus-Wise [27]). — We have ∗ ct g ct ι DR (A) = DR (A) ∈ A M . g g,n By deﬁnition, DR (A) is a Chow class on the moduli space of stable curves. In fact, the double ramiﬁcation cycle lies in the tautological ring. Proposition 2 (Faber-Pandharipande [11]). — DR (A) ∈ R (M ). g g,n The proof of [11] provides an algorithm to calculate DR (A) in the tautological ring, but the complexity of the method is too great: there is no apparent way to obtain an explicit formula for DR (A) directly from [11]. 0.3. Stable graphs and strata 0.3.1. Summation over stable graphs The strata of M are the quasi-projective subvarieties parameterizing pointed g,n curves of a ﬁxed topological type. The moduli space M is a disjoint union of ﬁnitely many g,n strata. The main result of the paper is a proof of an explicit formula conjectured by Pixton [34] in 2014 for DR (A) in the tautological ring R (M ). The formula is written in g g,n terms of a summation over stable graphs indexing the strata of M .Wereviewhere g,n the standard indexing of the strata of M by stable graphs. g,n 0.3.2. Stable graphs The strata of the moduli space of curves correspond to stable graphs = (V, H, L, g : V → Z ,v : H → V,ι : H → H) ≥0 satisfying the following properties: Compatibility on the smaller open set of curves with rational tails was proven earlier in [4]. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 227 (i) V is a vertex set with a genus function g : V → Z , ≥0 (ii) H is a half-edge set equipped with a vertex assignment v : H → Vand an involution ι, (iii) E, the edge set, is deﬁned by the 2-cycles of ι in H (self-edges at vertices are permitted), (iv) L, the set of legs, is deﬁned by the ﬁxed points of ι and is placed in bijective correspondence with a set of markings, (v) the pair (V, E) deﬁnes a connected graph, (vi) for each vertex v, the stability condition holds: 2g(v) − 2 + n(v) > 0, where n(v) is the valence of at v including both half-edges and legs. An automorphism of consists of automorphisms of the sets V and H which leave in- variant the structures L, g, v,and ι.Let Aut( ) denote the automorphism group of . The genus of a stable graph is deﬁned by: g( ) = g(v) + h ( ). v∈V A quasi-projective stratum of M corresponding to Deligne-Mumford stable curves of g,n ﬁxed topological type naturally determines a stable graph of genus g with n legs by con- sidering the dual graph of a generic pointed curve parameterized by the stratum. Let G be the set of isomorphism classes of stable graphs of genus g with n legs. g,n The set G is ﬁnite. g,n 0.3.3. Strata classes To each stable graph ∈ G , we associate the moduli space g,n M = M . g(v),n(v) v∈V There is a canonical morphism (6) ξ : M → M g,n with image equal to the closure of the stratum associated to the graph . To construct ξ , a family of stable pointed curves over M is required. Such a family is easily deﬁned by attaching the pull-backs of the universal families over each of the M along the g(v),n(v) sections corresponding to the two halves of each edge in E( ). The degree of ξ is |Aut( )|. 228 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Each half-edge h ∈ H( ) determines a cotangent line L → M . If h ∈ L( ),then L is the pull-back via ξ of the corresponding cotangent line of M . h g,n If h is a side of an edge e ∈ E( ),then L is the cotangent line of the corresponding side of a node. Let ψ = c (L ) ∈ A (M , Q). h 1 h 0.4. Pixton’s formula 0.4.1. Weightings Let A = (a ,..., a ) be a vector of double ramiﬁcation data. Let ∈ G be a 1 n g,n stable graph of genus g with n legs. A weighting of is a function on the set of half-edges, w : H( ) → Z, which satisﬁes the following three properties: (i) ∀h ∈ L( ), corresponding to the marking i ∈{1,..., n}, w(h ) = a , i i (ii) ∀e ∈ E( ), corresponding to two half-edges h, h ∈ H( ), w(h) + w h = 0, (iii) ∀v ∈ V( ), w(h) = 0, v(h)=v where the sum is taken over all n(v) half-edges incident to v. If the graph has cycles, may carry inﬁnitely many weightings. Let r be a positive integer. A weighting mod r of is a function, w : H( ) →{0,..., r − 1}, which satisﬁes exactly properties (i)–(iii) above, but with the equalities replaced, in each case, by the condition of congruence mod r. For example, for (i), we require w(h ) = a mod r. i i h ( ) Let W be the set of weightings mod r of . The set W is ﬁnite, with cardinality r . ,r ,r We view r as a regularization parameter. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 229 0.4.2. Pixton’s conjecture Let A = (a ,..., a ) be a vector of double ramiﬁcation data. Let r be a positive 1 n d,r d integer. We denote by P (A) ∈ R (M ) the degree d component of the tautological g,n class 1 1 ξ exp a ψ ∗ h 1 i i h ( ) |Aut( )| r ∈G w∈W i=1 g,n ,r 1 − exp(−w(h)w(h )(ψ + ψ )) h h × , ψ + ψ h h e=(h,h )∈E( ) in R (M ). g,n Inside the push-forward in the above formula, the ﬁrst product exp a ψ i i i=1 is over h ∈ L( ) via the correspondence of legs and markings. The second product is over all e ∈ E( ).The factor 1 − exp(−w(h)w(h )(ψ + ψ )) h h ψ + ψ h h is well-deﬁned since • the denominator formally divides the numerator, • the factor is symmetric in h and h . No edge orientation is necessary. d,r The following fundamental polynomiality property of P (A) has been proven by Pixton. Proposition 3 (Pixton [35]). — For ﬁxed g, A, and d , the class d,r d P (A) ∈ R (M ) g,n is polynomial in r (for all sufﬁciently large r). d d,r We denote by P (A) the value at r = 0 of the polynomial associated to P (A) by g g Proposition 3.Inother words, P (A) is the constant term of the associated polynomial in r. For the reader’s convenience, Pixton’s proof of Proposition 3 is given in the Appendix. The main result of the paper is a proof of the formula for double ramiﬁcation cycles conjectured earlier by Pixton [34]. 230 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Theorem 1. —For g ≥ 0 and double ramiﬁcation data A,wehave −g g g DR (A) = 2 P (A) ∈ R (M ). g g,n For d < g, the classes P (A) do not yet have a geometric interpretation. However for d > g, the following vanishing conjectured by Pixton is now established. Theorem 2 (Clader-Janda [8]). — For all g ≥ 0, double ramiﬁcation data A,and d > g, we have d d P (A) = 0 ∈ R (M ). g,n Pixton [34] has further proposed a twist of the formula for P (A) by k ∈ Z.The codimension g class in the k = 1 case has been (conjecturally) related to the moduli spaces of meromorphic differential in the Appendix of [13]. The vanishing result of Clader-Janda [8] proves Pixton’s vanishing conjecture in codimensions d > g for all k. The k-twisted theory will be discussed in Section 1. In a forthcoming paper [22], we will generalize Pixton’s formula to the situation of maps to a P -rubber bundle over a target manifold X. 0.5. Basic examples 0.5.1. Genus 0 Let g = 0and A = (a ,..., a ) be a vector of double ramiﬁcation data. The graphs 1 n ∈ G are trees. Each ∈ G has a unique weighting mod r (for each r). We can directly 0,n 0,n calculate: d 2 2 (7)P (A) = P (A) = exp a ψ − a δ , 0 i I,J 0 i I d≥0 i=1 I J={1,...,n} |I|,|J|≥2 where a = a . I i i∈I By Theorem 1, the double ramiﬁcation cycle is the degree 0 term of (7), −0 0 DR (A) = 2 P (A) = 1. The vanishing of Theorem 2 implies 2 2 1 a ψ − a δ = 0 ∈ R (M ), i I,J 0,n i I i=1 I J={1,...,n} |I|,|J|≥2 where δ ∈ R (M ) is the class of the boundary divisor with marking distribution I J. I,J 0,n The above vanishing can be proven here directly by expressing ψ classes in terms of boundary divisors. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 231 0.5.2. Genus 1 Let g = 1and A = (a ,..., a ) be a vector of double ramiﬁcation data. Denote by 1 n δ ∈ R (M ) the boundary divisor of singular stable curves with a nonseparating node. 0 1,n Our convention is δ = ξ [M ],ξ : M → M . 0 ∗ 0,n+2 0,n+2 1,n The division by the order of the automorphism group is included in the deﬁnition of δ . Denote by δ ∈ R (M ) I 1,n the class of the boundary divisor of curves with a rational component carrying the mark- ings I ⊂{1,..., n} and an elliptic component carrying the markings I . We can calculate directly from the deﬁnitions: 1 2 2 (8)P (A) = a ψ − a δ − δ . i I 0 1 i I i=1 I⊂{1,...,n} |I|≥2 As before, a = a .ByTheorem 1, I i i∈I DR (A) = P (A). In genus 1, the virtual class plays essentially no role (since the moduli spaces of maps to rubber are of expected dimension). The genus 1 formula was already known to Hain [19]. It is instructive to compute the coefﬁcient − of δ .The class δ corresponds to 0 0 the graph with one vertex of genus 0 and one loop, h ( ) = 1. According to the deﬁnition, the coefﬁcient of δ is the r-free term of the polynomial r−1 w(r − w) , w=1 which is −B =− . 6 232 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE 0.5.3. Degree 0 Let g ≥ 0and A = (0,..., 0),so μ = ν =∅. The canonical map from the moduli of stable maps to the moduli of curves is then an isomorphism ∼ ∼ : M P , ∅, ∅ −→ M . g,n g,n By an analysis of the obstruction theory, g g DR (0,..., 0) = (−1) λ ∈ R (M ) g g g,n where λ is the top Chern class of the Hodge bundle E → M . g,n Let ∈ G be a stable graph. By the deﬁnition of a weighting mod r and the condi- g,n tion a = 0 for all markings i, the weights w(h), w h on the two halves of every separating edge e of must both be 0. The factor in Pixton’s formula for e, 1 − exp(−w(h)w(h )(ψ + ψ )) h h ψ + ψ h h then vanishes and kills the contribution of to P (0,..., 0). Let g ≥ 1. Let ξ be the map to M associated to the divisor of curves with non- g,n separating nodes ξ : M → M . g−1,n+2 g,n Since graphs with separating edges do not contribute to P (0,..., 0) and the trivial graph with no edges does not contribute in codimension g, g−1 P (0,..., 0) = ξ (0,..., 0) g g−1 for an explicit tautological class g−1 g−1 (0,..., 0) ∈ R (M ). g−1,n+2 g−1 Corollary 3. —For g ≥ 1, we have g−1 −g g λ = (−2) ξ (0,..., 0) ∈ R (M ). g ∗ g,n g−1 DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 233 ct The class λ is easily proven to vanish on M via the Abel-Jacobi map (4)and g,n well-known properties of the moduli space of abelian varieties [38]. Hence, there exists a g−1 Chow class γ ∈ A (M ) satisfying g−1,n+2 g−1,n+2 λ = ξ γ ∈ A (M ). g ∗ g−1,n+2 g,n g−1 Corollary 3 is a much stronger statement since (0,..., 0) is a tautological class on g−1 M given by an explicit formula. No such expressions were known before. g−1,n+2 • For M , Corollary 3 yields 1,n 1−1 1 λ =− ξ (0,..., 0) ∈ R (M ). 1 ∗ 1,n 1−1 Moreover, by calculation (8), 1−1 (0,..., 0) =− [M ], 0,n+2 1−1 since δ already carries a for the graph automorphism. Hence, we recover the well- known formula λ = ξ [M ]∈ R (M ). 1 ∗ 0,n+2 1,n • For M , Corollary 3 yields 2−1 2 λ = ξ (∅) ∈ R (M ). 2 ∗ 2 2−1 Denote by α the class of the boundary stratum of genus 0 curves with two self- intersections. We include the division by the order of the automorphism group in the deﬁnition, α = ξ [M ]. ∗ 0,4 Denote by β the class of the boundary divisor of genus 1 curves with one self-intersection multiplied by the ψ -class at one of the branches. Then we have 1 1 2−1 (9) ξ (∅) = α + β. 2−1 36 60 The classes α and β form a basis of R (M ). The expression 1 1 λ = α + β 144 240 can be checked by intersection with the two boundary divisors of M . 2 234 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE The coefﬁcient of α in (9) is obtained from the r-free term of w w (r − w )(r − w ). 1 2 1 2 1≤w ,w ≤r−1 1 2 The answer is B = . The coefﬁcient of β is the r-free term of r−1 2 2 − w (r − w) 2r w=1 1 1 given by − B = . 2 60 0.6. Strategy of proof The proof of Proposition 2 given in [11] uses the Gromov-Witten theory of P relative to the point ∞∈ P . The localization relations of [11] intertwine Hodge classes on the moduli spaces of curves with cycles obtained from maps to rubber. The interplay is very complicated with many auxiliary classes (not just the double ramiﬁcation cycles). Our proof of Theorem 1 is obtained by studying the Gromov-Witten theory of 1 8 1 1 1 P with an orbifold point BZ at 0 ∈ P and a relative point ∞∈ P .Let (P [r], ∞) denote the resulting orbifold/relative geometry. The proof may be explained conceptu- ally as studying the localization relations for all (P [r], ∞) simultaneously.For large r,the localization relations (after appropriate rescaling) may be viewed as having polynomial coefﬁcients in r. Then the r = 0 specialization of the linear algebra exactly yields Theo- rem 1. The regularization parameter r in the deﬁnition of Pixton’s formula in Section 0.4 is related to the orbifold structure BZ . Our proof is inspired by the structure of Pixton’s formula—especially the existence of the regularization parameter r. The argument requires two main steps: • Let M be the moduli space of rth roots (with respect to the tensor product) of g;a ,...,a 1 n the bundles O ( a x ) associated to pointed curves [C, p ,..., p ].Let C i i 1 n i=1 π : C → M g;a ,...,a g;a ,...,a 1 n 1 n be the universal curve, and let L be the universal rth root over the universal curve. Chiodo’s formulas [6] allow us to prove that the push-forward of r · c (−R π L) to M g ∗ g,n −g g,r and 2 P (A) are polynomials in r with the same constant term. Pixton’s formula there- fore has a geometric interpretation in terms of the intersection theory of the moduli space of rth roots. 8 Z BZ is the quotient stack obtained by the trivial action of Z on a point. r r = rZ 9 1 The target (P [r], ∞) was previously studied in [24] in the context of Hurwitz-Hodge integrals. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 235 • We use the localization formula [15] for the virtual class of the moduli space of stable maps to the orbifold/relative geometry (P [r], ∞). The monodromy conditions at 0 are given by μ and the ramiﬁcation proﬁle over ∞ is given by ν . The push-forward of the localization formula to M is a Laurent series in the equivariant parameter t and in r. g,n −1 0 The coefﬁcient of t r must vanish by geometric considerations. We prove the relation −1 0 obtained from the coefﬁcient of t r has only two terms. The ﬁrst is the constant term in r of the push-forward of r · c (−R π L) to M . The second term is the double ramiﬁ- g ∗ g,n cation cycle DR (A) with a minus sign. The vanishing of the sum of the two terms yields Theorem 1. 1. Moduli of stable maps to BZ 1.1. Twisting by k Let k ∈ Z.Avector A = (a ,..., a ) of k-twisted double ramiﬁcation data for genus g ≥ 0 1 n is deﬁned by the condition (10) a = k(2g − 2 + n). i=1 For k = 0, the condition (10)does not depend upon on the genus g and specializes to the balancing condition of the double ramiﬁcation data of Section 0.2.1. d,k d We recall the deﬁnition [34]ofPixton’s k-twisted cycle P (A) ∈ R (M ).Incase g,n k = 0, d,0 d d P (A) = P (A) ∈ R (M ). g,n g g Let ∈ G be a stable graph of genus g with n legs. A k-weighting mod r of is a g,n function on the set of half-edges, w : H( ) →{0,..., r − 1}, which satisﬁes the following three properties: (i) ∀h ∈ L( ), corresponding to the marking i ∈{1,..., n}, w(h ) = a mod r, i i (ii) ∀e ∈ E( ), corresponding to two half-edges h, h ∈ H( ), w(h) + w h =0mod r We always assume the stability condition 2g − 2 + n > 0holds. 236 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE (iii) ∀v ∈ V( ), w(h) = k 2g(v) − 2 + n(v) mod r, v(h)=v where the sum is taken over all n(v) half-edges incident to v. We denote by W the set of all k-weightings mod r of . ,r,k Let A be a vector of k-twisted ramiﬁcation data for genus g. For each positive d,r,k d integer r,let P (A) ∈ R (M ) be the degree d component of the tautological class g,n 1 1 2 2 −k κ (v) a ψ 1 h ξ e e h ( ) |Aut( )| r ∈G w∈W v∈V( ) i=1 g,n ,r,k −w(h)w(h )(ψ +ψ ) 1 − e × , ψ + ψ h h e=(h,h )∈E( ) in R (M ). g,n All the conventions here are the same as in the deﬁnitions of Section 0.4.2.Anew factor appears: κ (v) is the ﬁrst κ class, κ (v) ∈ R (M ), 1 g(v),n(v) on the moduli space corresponding to a vertex v of the graph . As in the untwisted case, polynomiality holds (see the Appendix). Proposition 3 (Pixton [35]). — For ﬁxed g, A, k, and d , the class d,r,k d P (A) ∈ R (M ) g,n is polynomial in r (for all sufﬁciently large r). d,k d,r,k We denote by P (A) the value at r = 0 of the polynomial associated to P (A) g g d,k by Proposition 3 .Inother words, P (A) is the constant term of the associated polynomial in r. In case k = 0, Proposition 3 specializes to Proposition 3. Pixton’s proof is given in the Appendix. i+1 11 i Our convention is κ = π (ψ ) ∈ R (M ) where i ∗ g(v),n(v) n(v)+1 π : M → M g(v),n(v)+1 g(v),n(v) is the map forgetting the marking n(v) + 1. For a review of κ and cotangent ψ classes, see [16]. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 237 1.2. Generalized r-spin structures 1.2.1. Overview Let [C, p ,..., p ]∈ M be a nonsingular curve with distinct markings and 1 n g,n canonical bundle ω .Let ω = ω (p + ··· + p ) log C 1 n be the log canonical bundle. Let k and a ,..., a be integers for which k(2g − 2 + n) − a is divisible by a 1 n i positive integer r.Then rth roots L of the line bundle ⊗k ω − a p i i log on C exist.The spaceofsuch rth tensor roots possesses a natural compactiﬁcation r,k M constructed in [5, 23]. The moduli space of rth roots carries a universal curve g;a ,...,a 1 n r,k r,k π : C → M g;a ,...,a g;a ,...,a 1 n 1 n and a universal line bundle r,k L → C g;a ,...,a 1 n equipped with an isomorphism ⊗r ⊗k (11) L −→ ω − a x i i log r,0 r,k on C .Incase k = 0, M is the space of stable maps [1] to the orbifold BZ . g;a ,...,a g;a ,...,a 1 n 1 n r,k We review here the basic properties of the moduli spaces M which we re- g;a ,...,a 1 n quire and refer the reader to [5, 23] for a foundational development of the theory. 1.2.2. Covering the moduli of curves The forgetful map to the moduli of curves r,k : M → M g,n g;a ,...,a 1 n 2g is an r -sheeted covering ramiﬁed over the boundary divisors. The degree of equals 2g−1 r because each rth root L has a Z symmetry group obtained from multiplication by rth roots of unity in the ﬁbers. The Z -action respects the isomorphism (11). r 238 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE To transform to an unramiﬁed covering in the orbifold sense, the orbifold structure of M must be altered. We introduce an extra Z stabilizer for each node of each stable g,n r curve, see [5]. The new orbifold thus obtained is called the moduli space of r-stable curves. 1.2.3. Boundary strata The rth roots of the trivial line bundle over the cylinder are classiﬁed by a pair of integers (12)0 ≤ a , a ≤ r − 1, a + a =0mod r attached to the borders of the cylinder. Therefore, the nodes of stable curves in the com- r,k pactiﬁcation M carry the data (12)with a and a assigned to the two branches g;a ,...,a 1 n meeting at the node. In case thenodeis separating,the data (12) are determined uniquely by the way the genus and the marked points are distributed over the two components of the curve. In case thenodeis nonseparating, the data can be arbitrary and describes different boundary divisors. r,k More generally, the boundary strata of M are described by stable graphs g;a ,...,a 1 n ∈ G together with a k-weighting mod r.The data (12) is obtained from the weighting g,n w of the half-edges of . 1.2.4. Normal bundles of boundary divisors The normal bundle of a boundary divisor in M has ﬁrst Chern class g,n − ψ + ψ , where ψ and ψ are the ψ -classes associated to the branches of the node. In the moduli r,k space of r-stable curves and in the space M , the normal bundle to a boundary g;a ,...,a 1 n divisor has ﬁrst Chern class ψ + ψ (13) − , where the classes ψ and ψ are pulled back from M . g,n The two standard boundary maps M → M g−1,n+2 g,n and M × M → M g ,n +1 g ,n +1 g,n 1 1 2 2 More precisely, the normal bundle of the associated gluing morphism ξ . DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 239 are, in the space of rth roots, replaced by diagrams r,k φ r,k nonsep (14) M ←− B −→ M g−1;a ,...,a ,a ,a a ,a g;a ,...,a 1 n 1 n and r,k r,k ϕ j r,k sep (15) M × M ←− B −→ M . g ;a ,...,a ,a g ;a ,a ,...,a a ,a g;a ,...,a 1 1 n 2 n +1 n 1 n 1 1 The maps φ and ϕ are of degree 1, but in general are not isomorphisms, see [7], Sec- tion 2.3. Remark 1. — There are two natural ways to introduce the orbifold structure on the space of rth roots. In the ﬁrst, every node of the curve contributes an extra Z/rZ in the stabilizer. In the second, a node of type (a , a ) contributes an extra Z/gcd(a , r)Z to the stabilizer. The ﬁrst is used in [5], the second in [1]. Here, we follow the ﬁrst convention to avoid gcd factors appearing the computations. In Remark 2, we indicate the changes which must be made in the computations if the second convention is followed. The ﬁnal result is the same (independent of the choice of convention). 1.2.5. Intersecting boundary strata Let ( ,w ) and ( ,w ) be two stable graphs in G equipped with k-weightings A A B B g,n r,k mod r. The intersection of the two corresponding boundary strata of M can be g;a ,...,a 1 n determined as follows. Enumerate all pairs ( , w) of stable graphs with k-weightings mod r whose edges are marked with letters A, B or both in such a way that contracting all edges outside A yields ( ,w ) and contracting all edges outside B yields ( ,w ). A A B B Assign a factor − ψ + ψ to every edge that is marked by both A and B. The sum of boundary strata given by these graphs multiplied by the appropriate combinations of ψ -classes represents the intersec- tion of the boundary components ( ,w ) and ( ,w ). A A B B 1.3. Chiodo’s formula Let g be the genus, and let k ∈ Z and a ,..., a ∈ Z satisfy 1 n k(2g − 2 + n) − a =0mod r for an integer r > 0. See the Appendix of [16] for a detailed discussion in the case of M . g,n 240 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Following the notation of Section 1.2,let r,k r,k π : C → M g;a ,...,a g;a ,...,a 1 n 1 n r,k be the universal curve and L → C the universal rth root. Our aim here is to de- g;a ,...,a 1 n scribe the total Chern class r,k ∗ ∗ c −R π L ∈ A M g;a ,...,a 1 n following the work of A. Chiodo [6]. Denote by B (x) the m-th Bernoulli polynomial, n xt t te B (x) = . n! e − 1 n=0 Proposition 4. — The total Chern class c(−R π L) is equal to B (k/r) 1 m−1 m+1 |E( )| − (−1) κ (v) m≥1 m(m+1) r (ξ ) e ,w ∗ |Aut( )| ∈G w∈W v∈V( ) g,n ,r,k B (a /r) m−1 m+1 i m (−1) ψ m≥1 m(m+1) h × e i=1 B (w(h)/r) m−1 m+1 m m (−1) [(ψ ) −(−ψ ) ] m≥1 h m(m+1) 1 − e ψ + ψ h h e∈E( ) e=(h,h ) r,k in A (M ). g;a ,...,a 1 n Here, ξ is the map of the boundary stratum corresponding to ( , w) to the ,w r,k moduli space M , g;a ,...,a 1 n r,k ξ : M → M . ,w ,w g;a ,...,a 1 n Every factor in the last product is symmetric in h and h since w r − w m+1 B = (−1) B m+1 m+1 r r for m ≥ 1. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 241 Proof. — The proposition follows from Chiodo’s formula [6] for the Chern charac- ters of R π L, k a B ( ) B ( ) m+1 m+1 r r m (16) ch (r, k; a ,..., a ) = κ − ψ m 1 n m (m + 1)! (m + 1)! i=1 r−1 m m B ( ) r (ψ ) − (−ψ ) m+1 + ξ , w∗ 2 (m + 1)! ψ + ψ w=0 and from the universal relation • m • • b c −E = exp (−1) (m − 1)!ch E , E ∈ D . m≥1 |E( )| The factor r in the proposition is due to the factor of r in the last term of Chiodo’s formula. When two such terms corresponding to the same edge are multi- plied, the factor r partly cancels with the factor in the ﬁrst Chern class of the normal bundle − (ψ + ψ ). Thus only a single factor of r per edge remains. ∗ ∗ Corollary 4. — The push-forward c(−R π L) ∈ R (M ) is equal to ∗ ∗ g,n 2g−1−h ( ) B (k/r) r m−1 m+1 − (−1) κ (v) m≥1 m(m+1) ξ e |Aut( )| ∈G w∈W v∈V( ) g,n ,r,k B (a /r) m−1 m+1 i m (−1) ψ m≥1 m(m+1) h × e i=1 B (w(h)/r) m−1 m+1 m m (−1) [(ψ ) −(−ψ ) ] m≥1 h m(m+1) 1 − e × . ψ + ψ h h e∈E( ) e=(h,h ) Proof. — Because the maps φ and ϕ of Equations (14)and (15)havedegree1,the degree of the map r,k (17) : M → M g,n g;a ,...,a 1 n restricted to the stratum associated to ( , w) is equal to the product of degrees over vertices, that is, (2g −1) v∈V( ) r . r 1 The factor in the last term is ,but the accounts for the degree 2 of the boundary map ξ obtained from the 2 2 ordering of the nodes. 242 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE The formula of the corollary is obtained by push-forward from the formula of Proposi- tion 4.The powerof r is given by |E|+ (2g − 1) =|E|−|V|+ 2 g v v v∈V( ) v∈V( ) 1 1 = h ( ) − 1 + 2 g − h ( ) = 2g − 1 − h ( ). 1.4. Reduction modulo r Let g be the genus, and let k ∈ Z and a ,..., a ∈ Z satisfy 1 n k(2g − 2 + n) − a =0mod r for an integer r > 0. Let A = (a ,..., a ) 1 n be the associated vector of k-twisted double ramiﬁcation data. Proposition 3 (Pixton [35]). — Let ∈ G be a ﬁxed stable graph. Let Q be a polynomial g,n with Q-coefﬁcients in variables corresponding to the set of half-edges H( ). Then the sum F(r) = Q w(h ), ...,w(h ) 1 |H( )| w∈W ,r,k h ( ) over all possible k-weightings mod r is (for all sufﬁciently large r) a polynomial in r divisible by r . Proposition 3 implies Proposition 3 (which implies Proposition 3 under the spe- cialization k = 0). Proposition 3 is the basic statement. See the Appendix for the proof. We will require Proposition 3 to prove the following result. Let r,k : M → M g,n g;a ,...,a 1 n be the map to the moduli of curves. Let d ≥ 0 be a degree and let c denote the d -th Chern class. Proposition 5. — The two cycle classes 2d−2g+1 ∗ d −d d,r,k d r c −R π L ∈ R (M ) and 2 P (A) ∈ R (M ) ∗ d ∗ g,n g,n are polynomials in r (for r sufﬁciently large). Moreover, the two polynomials have the same constant term. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 243 Proof. — Choose a stable graph decorated with κ classes on the vertices and ψ classes on half-edges, [ , γ ],γ = M κ(v) · ψ . v∈V( ) h∈H( ) Here, M is a monomial in the κ classes at v and m is a non-negative integer for each h. v h The decorated graph data [ , γ ] represents a tautological class of degree d.Wewill study the coefﬁcient of the decorated graph in the formula for c (−R π L) given by ∗ d ∗ Corollary 4 for r sufﬁciently large. According to Proposition 3 , the coefﬁcient of [ , γ ] in the formula for the push- forward c (−R π L) is a Laurent polynomial in r. Indeed, the formula of Corollary 4 ∗ d ∗ is a combination of a ﬁnite number of sums of the type of Proposition 3 multiplied by positive and negative powers of r. Each contributing sum is obtained by expanding the exponentials in the formula of Corollary 4 and selecting a degree d monomial. To begin with, let us determine the terms of lowest power in r in the Laurent poly- nomial obtained by summation in the formula of Corollary 4.The (m + 1)st Bernoulli polynomial B has degree m + 1. Therefore, in the formula of Corollary 4, every class m+1 m 16 m−1 of degree m,beit κ , ψ or an edge with a class ψ , appears with a power of r 1 17 equal to or larger. If we take a product of q coefﬁcients of expanded exponentials of m+1 Bernoulli polynomials, we will obtain a power of r equal to or larger. Thus the lowest d+q possible power of r is obtained if we take a product of d classes of degree 1 each. Then, the power of r is equal to which is the lowest possible value. 2d After we perform the summation over all k-weightings mod r of the graph ,we h ( ) obtain a polynomial in r divisible by r by Proposition 3 . The lowest possible power h ( )−2d of r after summation becomes r . Finally, formula of Corollary 4 carries a global 2g−1−h ( ) ∗ factor of r . We conclude c (−R π L) is a Laurent polynomial in r with lowest ∗ d ∗ 2g−2d−1 power of r equal to r . In other words, the product 2d−2g+1 ∗ r c −R π L ∗ d ∗ is a polynomial in r. We now identify more precisely the terms of lowest power in r. As we have seen above, the lowest possible power of r in the formula of Corollary 4 is obtained by choosing only the terms that contain a degree 1 class multiplied by the second Bernoulli polyno- mial, w w w 1 B = − + . r r r 6 The edge contributes 1 to the degree of the class. 17 1 The power of is −m − 1. By larger here, we mean larger power (so smaller pole or polynomial terms). m+1 r 244 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Moreover, in the Bernoulli polynomial only the ﬁrst term will contribute to the lowest power of r. After dropping all the terms which do not contribute to the lowest power of r, the formula of Corollary 4 simpliﬁes to 1 n 2g−1−h ( ) 2 r k a ξ exp − κ (v) exp ψ ∗ 1 h 2 2 |Aut( )| 2r 2r ∈G w∈W v∈V( ) i=1 g,n ,r,k w(h) 1 − exp[ (ψ + ψ )] 2 h h 2r × . ψ + ψ h h e=(h,h )∈E( ) We transform w(h) in the last factor to −w(h)w(h ) for symmetry. Indeed, w(h) = w(h) r − w h =−w(h)w h mod r, so the lowest degree in r is not affected. After factoring out the 2r in the denominators, we obtain −h ( ) 2 2 2g−1−2d −d −k κ (v) a ψ 1 h r · 2 · ξ e e |Aut( )| ∈G w∈W v∈V( ) i=1 g,n ,r,k −w(h)w(h )(ψ +ψ ) 1 − e × . ψ + ψ h h e=(h,h )∈E( ) 2d−2g+1 −d d,k,r Multiplication by r then yields exactly 2 P (A) as claimed. 2. Localization analysis 2.1. Overview Let A = (a ,..., a ) be a vector of double ramiﬁcation data as deﬁned in Sec- 1 n tion 0.2.1, a = 0. i=1 From the positive and negative parts of A, we obtain two partitions μ and ν of the same size |μ|=|ν|. The double ramiﬁcation cycle DR (A) ∈ R (M ) g g,n 1 ∼ is deﬁned via the moduli space of stable maps to rubber M (P ,μ,ν) where I corre- g,I sponds to the 0 parts of A. We prove here the claim of Theorem 1, −g g g DR (A) = 2 P (A) ∈ R (M ). g g,n g DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 245 2.2. Target geometry Following the notation of [24], let P [r] be the projective line with an orbifold point BZ at 0 ∈ P .Let M P [r],ν g,I,μ be the moduli space of stable maps to the orbifold/relative pair (P [r], ∞). The moduli space parameterizes connected, semistable, twisted curves C of genus g with I markings together with a map f : C → P 1 1 where P is a destabilization of P [r] over ∞∈ P [r]. We refer the reader to [1, 24] for the deﬁnitions of the moduli space of stable maps 1 1 to (P [r], ∞). The following conditions are required to hold over 0, ∞∈ P [r]: • The stack structure of the domain curve C occurs only at the nodes over 0 ∈ P [r] and the markings corresponding to μ (which must to be mapped to 0 ∈ P [r]). The monodromies associated to the latter markings are speciﬁed by the parts μ of μ. For each part μ ,let i i μ = μ mod r where 0 ≤ μ ≤ r − 1. i i • The map f is ﬁnite over ∞∈ P [r] on the last component of P with ramiﬁcation data given by ν.The (ν) ramiﬁcation points are marked. The map f satisﬁes the ramiﬁcation matching condition over the internal nodes of the destabiliza- tion P. The moduli space M (P [r],ν) has a perfect obstruction theory and a virtual g,I,μ class of dimension (μ) |ν| μ vir (18)dim M P [r],ν = 2g − 2 + n + − , C g,I,μ r r i=1 see [24, Section 1.1]. Recall n is the length of A, n = (μ) + (ν) + (I). Since |μ|=|ν|, the virtual dimension is an integer. We will be most interested in the case where r > |μ|=|ν|. Then, μ = μ for all the parts of μ, and formula (18) for the virtual dimension simpliﬁes to vir dim M P [r],ν = 2g − 2 + n. C g,I,μ 246 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE 2.3. C -ﬁxed loci ∗ 1 The standard C -action on P deﬁned by ξ ·[z , z ]=[z ,ξ z ] 0 1 0 1 ∗ 1 1 lifts canonically to C -actions on P [r] and M (P [r],ν). We describe here the g,I,μ ∗ 1 C -ﬁxed loci of M (P [r],ν). g,I,μ ∗ 1 The C -ﬁxed loci of M (P [r],ν) are labeled by decorated graphs .The ver- g,I,μ tices v ∈ V() are decorated with a genus g(v) and the legs are labeled with markings in μ ∪ I ∪ ν.Eachvertex v is labeled by 1 1 0 ∈ P [r] or ∞∈ P [r]. Each edge e ∈ E() is decorated with a degree d that corresponds to the d -th power e e map 1 1 P [r]→ P [r]. The vertex labeling endows the graph with a bipartite structure. Avertexof over 0 ∈ P [r] corresponds to a stable map contracted to 0 given by an element of M (BZ ) with monodromies μ(v) speciﬁed by the corresponding g(v),I(v),μ(v) r entries of μ and the negatives of the degrees d of the incident edges (modulo r). We will use the notation r r M = M = M (BZ ). g(v),I(v),μ(v) r v g(v);I(v),μ(v) r r The fundamental class [M ] of M is of dimension v v dim M = 3g(v) − 3 + I(v) + μ(v) . As explained in Section 1.2.2, : M → M g(v),n(v) is a ﬁnite covering of the moduli space M where g(v),n(v) n(v) = μ(v) + I(v) is the total number of markings and edges incident to v. The stable map over ∞∈ P [r] can take two different forms. If the target does not expand, then the stable map has (ν) preimages of ∞∈ P [r]. Each preimage is described by an unstable vertex of labeled by ∞. If the target expands, then the stable map is a possibly disconnected rubber map. The ramiﬁcation data is given by incident edges over 0 of the rubber and by elements of ν over ∞ of the rubber. In this case every vertex v over ∞∈ P [r] describes a connected component of the rubber map. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 247 Let g(∞) be the genus of the possibly disconnected domain of the rubber map. We will denote the moduli space of stable maps to rubber by M and the virtual vir fundamental class by [M ] , vir dim M = 2g(∞) − 3 + n(∞), where n(∞) is the total number of markings and edges incident to vertices over ∞, n(∞) = I(∞) + (ν) +|E()|. vir The image of the virtual fundamental class [M ] in the moduli space of (not necessar- ily connected ) stable curves is denoted by DR . Avertex v ∈ V() is unstable when 2g(v) − 2 + n(v) ≤ 0. There are four types of unstable vertices: (i) v → 0, g(v) = 0, v carries no markings and one incident edge, (ii) v → 0, g(v) = 0, v carries no markings and two incident edges, (iii) v → 0, g(v) = 0, v carries one marking and one incident edge, (iv) v → ∞,g(v) = 0, v carries one marking and one incident edge. The target of the stable map expands iff there is at least one stable vertex over ∞. A stable map in the C -ﬁxed locus corresponding to is obtained by gluing to- gether maps associated to the vertices v ∈ V() with Galois covers associated to the 0 ∗ edges. Denote by V () the set of stable vertices of over 0. Then the C -ﬁxed locus st corresponding to is isomorphic to the product r ∼ 0 M × M , if the target expands, v∈V () v ∞ st M = 0 M , if the target does not expand, v∈V () v st quotiented by the automorphism group of and the product of cyclic groups Z associ- ated to the Galois covers of the edges. Thus the natural morphism corresponding to , ι : M → M P [r],ν , g,I,μ is of degree Aut() d , e∈E() onto the image ι(M ). Recall that D =|μ|=|ν| is thedegreeofthe stable map. We omit the data g(∞),I(∞), ν,and δ(∞) in the notation. By [3, Lemma 2.3], a double ramiﬁcation cycle vanishes as soon as there are at least two nontrivial components of the domain. However, we will not require the vanishing result here. Our analysis will naturally avoid disconnected domains mapping to the rubber over ∞∈ P [r]. 248 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Lemma 6. —For r > D, the unstable vertices of type (i) and (ii) can not occur. Proof. — At each stable vertex v ∈ V() over 0 ∈ P [r], the condition |μ |=0mod r has to hold in order for M (BZ ) to be non-empty. Because of the balancing con- g ,I ,μ r v v v dition at the nodes, the parts of μ at v have to add up to the sum of the degrees of the incident edges modulo r. Since r > D, the parts of μ at v must addupto exactly the sum of the degrees of the incident edges. So the incident edges of the stable vertices and the type (iii) unstable vertices must account for the entire degree D. Thus, there is no remaining edge degree for unstable vertices of type (i) and (ii). 2.4. Localization formula We write the C -equivariant Chow ring of a point as A (•) = Q[t], where t is the ﬁrst Chern class of the standard representation. For the localization formula, we will require the inverse of the C -equivariant Euler 1 ∗ class of the virtual normal bundle in M (P [r],ν) to the C -ﬁxed locus corresponding g,I,μ to .Let f : C → P [r], [f ]∈ M . The inverse Euler class is represented by 1 ∗ 1 e(H (C, f T 1 (−∞))) 1 1 P [r] (19) = . vir 0 ∗ e(H (C, f T 1 (−∞))) e(N ) e(N ) e(Norm ) P [r] i ∞ Formula (19) has several terms which require explanation. We assume r > |μ|=|ν|, so unstable vertices over 0 ∈ P [r] of type (i) and (ii) do not occur in . First consider the leading factor of (19), 1 ∗ e(H (C, f T (−∞))) P [r] (20) . 0 ∗ e(H (C, f T 1 (−∞))) P [r] To compute this factor one uses the normalization exact sequence for the domain C tensored with the line bundle f T 1 (−∞). The associated long exact sequence in co- P [r] homology decomposes the leading factor into a product of vertex, edge, and node con- tributions. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 249 • Let v ∈ V() be a stable vertex over 0 ∈ P [r] corresponding to a contracted component C . From the resulting stable map to the orbifold point C → BZ ⊂ P [r], v r we obtain an orbifold line bundle L = f T 1 | P [r] 0 on C which is an rth root of O . Therefore, the contribution v C 1 ∗ e(H (C , f T 1 | )) v P [r] 0 0 ∗ e(H (C , f T | )) v P [r] 0 yields the class ∗ (1/r) ∗ c −R π L ⊗ O ∈ A M ⊗ Q t, , rk ∗ g(v),I(v),μ(v) (1/r) where L → M is the universal rth root, O is a trivial line bundle g(v),I(v),μ(v) ∗ 1 with a C -action of weight ,and rk = g − 1 + E(v) is the virtual rank of −R π L. Unstable vertices of type (iii) over 0 ∈ P [r] and the vertices over ∞ contribute factors of 1. i ∗ • The edge contribution is trivial since the degree of f T 1 (−∞) is less than 1, P [r] see [24, Section 2.2]. • The contribution of a node N over 0 ∈ P [r] is trivial. The space of sec- 0 ∗ 1 tions H (N, f T 1 (−∞)) vanishes because N must be stacky, and H (N, P [r] f T 1 (−∞)) is trivial for dimension reasons. P [r] Nodes over ∞∈ P [r] contribute 1. Consider next the last two factors of (19), 1 1 e(N ) e(N ) i ∞ −1 1 • The product e(N ) is over all nodes over 0 ∈ P [r] of the domain C which are forced by the graph . These are nodes where the edges of are attached to the vertices. If N is a node over 0 forced by the edge e ∈ E() and the associated vertex v is stable, then t ψ (21) e(N) = − . rd r e 250 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE This expression corresponds to the smoothing of the node N of the domain curve: e(N) is the ﬁrst Chern classe of the normal bundle of the divisor of nodal domain curves. The ﬁrst Chern classes of the cotangent lines to the branches at the node are divided by r because of the orbifold twist, see Section 1.2. In the case of an unstable vertex of type (iii), the associated edge does not produce a node of the domain. The type (iii) edge incidences do not appear in −1 e(N ) . • N corresponds to the target degeneration over ∞∈ P [r].The factor e(N ) is ∞ ∞ 1ifthe target (P [r], ∞) does not degenerate and t + ψ e(N ) =− e∈E() if the target does degenerate [17]. Here, ψ is the ﬁrst Chern class of the cotan- gent line bundle to the bubble in the target curve at the attachment point to ∞ of P [r]. Summing up our analysis, we write out the outcome of the virtual localization formula [15]in A (M (P [r],ν)) ⊗ Q[t, ].Wehave ∗ g,I,μ vir 1 1 [M ] vir (22) M P [r],ν = · ι , g,I,μ ∗ vir |Aut()| d e(Norm ) e∈E() vir [M ] where is the product of the following factors: vir e(Norm ) r t ∗ g(v)−1+|E(v)|−d • · c (−R π L)( ) for each stable vertex v ∈ V() d ∗ e∈E(v) d≥0 −ψ r over 0, e∈E() •− if the target degenerates. t+ψ Remark 2. —Aswementioned in Remark 1, there are two conventions for the orbifold structure on the space of rth roots. In the paper, we follow the convention which assigns to each node of the curve the stabilizer Z/rZ. It is also possible to assign to a node of type (a , a ) the stabilizer Z/qZ,where q is the order of a and a in Z/rZ.Welisthere the required modiﬁcations of our formulas when the latter convention is followed. Denote p = gcd(r, a ) = gcd(r, a ).Wethenhave pq = r.InSection 1.2.4,the ﬁrst Chern class of the normal line bundle to a boundary divisor in the space of rth roots becomes −(ψ + ψ )/q rather than −(ψ + ψ )/r. The degrees of the maps φ and ϕ in Equations (14)and (15)are equalto p instead of 1. The factor in the last term of |E( )| Chiodo’s formula (16) becomes . Because of the above, the factor r in Proposition 4 |E( )| is transformed into q . The statement of Corollary 4 is unchanged, but in the proof |E( )| |E( )| |E( )| the factor r is now obtained as a product of q from Proposition 4 and p from the degree of φ and ϕ. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 251 In the localization formula, the Euler classes of the normal bundles at the nodes ψ ψ t e t e over 0 become − instead of − . On the other hand, at each node there are p qd q rd r e e ways to reconstruct an rth root bundle from its restrictions to the two branches. Thus the contribution of each node still has an r in the numerator. e(N) 2.5. Extracting the double ramiﬁcation cycle 2.5.1. Three operations We will now perform three operations on the localization formula (22)for the 1 vir virtual class [M (P [r],ν)] : g,I,μ (i) the C -equivariant push-forward via (23) : M P [r],ν → M g,I,μ g,n to the moduli space M with trivial C -action, g,n −1 (ii) extraction of the coefﬁcient of t after push-forward by , (iii) extraction of the coefﬁcient of r . −1 After push-forward by , the coefﬁcient of t is equal to 0 because vir 1 ∗ M P [r],ν ∈ A (M ) ⊗ Q[t]. ∗ g,I,μ g,n −1 Using the result of Section 1.4,all termsofthe t coefﬁcient will be seen to be polynomi- als in r, so operation (iii) will be well-deﬁned. After operations (i)–(iii), only two nonzero terms will remain. The cancellation of the two remaining terms will prove Theorem 1. To perform (i)–(iii), we multiply the -push-forward of the localization formula (22) 0 0 by t and extract the coefﬁcient of t r . To simplify the computations, we introduce the new variable s = tr. 0 0 0 0 Then, instead of extracting the coefﬁcient of t r ,weextract thecoefﬁcientof s r . 2.5.2. Push-forward to M g,n For each vertex v ∈ V(),wehave r r M = M . v g(v);I(v),μ(v) Following Section 1.2.2, we denote the map to moduli by (24) : M → M . g(v),n(v) g(v);I(v),μ(v) 252 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE The use of the notation in (23)and (24) is compatible. Denote by 2d−2g(v)+1 ∗ d (25) c = r c −R π L ∈ R (M ). d ∗ d ∗ g(v),n(v) By Proposition 5, c is a polynomial in r for r sufﬁciently large. From (22) and the contribution calculus for presented in Section 2.5.2,wehave a complete formula for the C -equivariant push-forward of t times the virtual class: vir (26) t M P [r],ν ∗ g,I,μ vir s 1 1 [M ] = · · ι , ∗ ∗ vir r |Aut()| d e(Norm ) e∈E() vir [M ] where ι is the product of the following factors: ∗ ∗ vir e(Norm ) • afactor r d 1 g(v)−d d · · c s ∈ R (M ) ⊗ Q s, d g(v),n(v) rd s 1 − ψ s e∈E(v) s d≥0 for each stable vertex v ∈ V() over 0, • afactor e∈E() − · · DR s 1 + ψ if the target degenerates. 2.5.3. Extracting coefﬁcients Extracting the coefﬁcient of r .— By Proposition 5, the classes c are polynomial in r for r sufﬁciently large. We have an r in the denominator in the prefactor on the right side of (26) which comes from the multiplication by t on the left side. However, in all other factors, we only have positive powers of r, with at least one r per stable vertex of the graph over 0 and one more r if the target degenerates. The only graphs which contribute to the coefﬁcient of r are those with exactly one r in the numerator. There are only two graphs which have exactly one r factor in the numerator: • the graph with a stable vertex of full genus g over 0 and (ν) type (iv) unstable vertices over ∞, • the graph with a stable vertex of full genus g over ∞ and (μ) type (iii) unstable vertices over 0. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 253 No terms involving ψ classes contribute to the r coefﬁcient of either or since every ψ class in the localization formula comes with an extra factor of r.Wecan now write the r coefﬁcient of the right side of (26): vir 1 g−d (27)Coeff 0 t M P [r],ν = Coeff 0 c s − DR (A) r ∗ g,I,μ r d g d≥0 d 1 in R (M ) ⊗ Q[s, ]. g,n Extracting the coefﬁcient of s .— The remaining powers of s in (27) appear only with the classes c in the contribution of the graph . In order to obtain s ,wetaketotake d = g, vir 0 0 0 (28)Coeff t M P [r],ν = Coeff [ c ]− DR (A) ∗ g,I,μ g g s r r in R (M ). g,n 2.5.4. Final relation 1 vir 0 0 Since Coeff [ (t[M (P [r],ν)] )] vanishes, we can rewrite (28)as ∗ g,I,μ s r 2g−2g+1 ∗ g (29)DR (A) = Coeff r c −R π L ∈ R (M ), g r ∗ g ∗ g,n using deﬁnition (25)of c .ByProposition 5 applied to the right side of relation (29), −g g g DR (A) = 2 P (A) ∈ R (M ). g g,n The proof of Theorem 1 is complete. 3. A few applications 3.1. A new expression for λ We discuss here a few more examples of the formula for Corollary 3: a formula for the class λ supported on the boundary divisor with a nonseparating node. We give the formulas for the class λ in M up to genus 4. g g The answer is expressed as a sum of labeled stable graphs with coefﬁcients where each stable graph has at least one nontrivial cycle. The genus of each vertex is written inside the vertex. The powers of ψ -classes are written on the corresponding half-edges (zero powers are omitted). The expressions do not involve κ -classes. Each labeled graph describes a moduli space M (a product of moduli spaces associated with its vertices), a tautological class α ∈ R (M ) and a natural map π : M → M . Our convention is that then represents the cycle class π α. For instance, if g ∗ the graph carries no ψ -classes, the class α equals 1 and the map π is of degree |Aut( )|. The cycle class represented by is then |Aut( )| times the class of the image of π . 254 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Genus 1. λ = . Genus 2. 1 1 λ = + . 2 1 0 240 1152 Genus 3. 1 1 1 1 ψ ψ λ = + − + 3 2 2 1 1 1 2016 2016 672 5760 13 1 1 − − + . 0 1 0 1 0 30240 5760 82944 Genus 4. 1 1 1 1 3 2 ψ ψ 2 λ = + − − 4 3 3 1 2 1 2 11520 3840 2880 3840 1 1 1 1 ψ ψ ψ ψ − − − − 1 2 1 2 1 2 1 2 ψ ψ 1440 1920 2880 3840 1 1 1 1 ψ ψ ψ + ψ + + + 2 2 2 1 ψ 1 48384 48384 115200 960 23 1 1 − − − 2 0 2 0 2 0 ψ ψ 100800 57600 16128 1 1 1 − − − 2 0 1 1 1 1 ψ ψ 16128 57600 16128 1 23 23 − − + 1 1 1 1 0 2 ψ ψ 16128 100800 100800 0 1 23 1 1 + + + 1 1 1 0 1 0 1 50400 16128 115200 1 13 1 + − − 1 1 0 1 0 276480 725760 138240 43 13 1 − − − 1 0 1 0 1 0 1612800 725760 276480 7962624 DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 255 All these expressions are obtained by taking n = 0 in the formula for the double ramiﬁcation cycle. 3.2. Hodge integrals Corollary 3 may be applied to any Hodge integral which contains the top Chern class of the Hodge bundle. As an example, we consider the following Hodge integral (ﬁrst calculated in [10]) related to the constant map contribution in the Gromov-Witten theory of Calabi-Yau threefolds. Proposition 7. —For g ≥ 1, we have 1 1 B B 2g 2g+2 λ λ λ =− . g+1 g g−1 2 (2g)! 2g 2g + 2 g+1 Proof. — We compute the integral by replacing λ with the expression obtained g+1 from Corollary 3. g+1 In Pixton’s formula for DR (∅) = (−1) λ , we only have to consider graphs g+1 g+1 with no separating edges. On the other hand, λ λ vanishes on strata as soon as there g g−1 are at least two nonseparating edges. Thus we are left with just one graph: the graph with a vertex of genus g and one loop. g+1 The edge term in Pixton’s formula for (−1) DR (∅) = λ for the unique g+1 g+1 remaining graph is (ψ +ψ ) 2k+2 1 − e a g+1 g (−1) = (−1) ψ + ψ . k+1 ψ + ψ 2 (k + 1)! k≥0 The contribution of the graph is therefore the r-free term of r−1 g 2g+2 (−1) a · (ψ + ψ ) λ λ . 1 2 g g−1 g+1 2r 2 (g + 1)! g,2 a=0 1 1 The factor of comes from the automorphism group of the graph, and the factor of 2 r comes from the ﬁrst Betti number of graph. Since r−1 2g+2 2 a = B r + O r , 2g+2 a=0 we see that the r-free term equals 2g+2 g g (−1) (ψ + ψ ) λ λ . 1 2 g g−1 g+2 2 (g + 1)! g,2 256 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Using Lemma 8 below we can rewrite this expression as B B 1 1 1 B B 2g+2 2g 2g 2g+2 g g+1 (−1) · (−1) =− , g+2 2 (g + 1)! 2g (2g − 1)!! 2 (2g)! 2g 2g + 2 which completes the derivation. Lemma 8. —For g ≥ 1, we have B 1 2g g g+1 (ψ + ψ ) λ λ = (−1) . 1 2 g g−1 2g (2g − 1)!! g,2 Proof. — According to Faber’s socle formula [9, 14], we have g+1 (−1) B p q 2g (30) ψ ψ λ λ = g g−1 1 2 2g 2 g(2p − 1)!!(2q − 1)!! g,2 whenever p + q = g. The formula holds even if p or q is equal to 0. After expanding (ψ + ψ ) and applying (30), we obtain 1 2 g! 1 2g g g+1 (ψ + ψ ) λ λ = (−1) . 1 2 g g−1 2g g 2 p!(2p − 1)!!q!(2q − 1)!! g,2 p+q=g The right side is, up to a factor, the sum of even binomial coefﬁcients in the 2g-th line of the Pascal triangle: g g 1 2 2 2g = = p!(2p − 1)!!q!(2q − 1)!! (2p)!(2q)! (2g)! 2p p+q=g p+q=g p=0 3g−1 = . (2g)! Finally, we have 3g−1 B g! 2 B 1 2g 2g g+1 g+1 (−1) · = (−1) 2g g 2 (2g)! 2g (2g − 1)!! as claimed. The socle formula is true if at most one power of a ψ class vanishes while the powers of all other ψ classes are positive. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 257 3.3. Local theory of curves The local Gromov-Witten theory of curves as developed in [2] starts with a fun- damental pairing with a double ramiﬁcation cycle (the resulting matrix then also appears in the quantum cohomology of the Hilbert scheme of points of C [30]). We rederive the double ramiﬁcation cycle pairing as an almost immediate consequence of Theorem 1. Proposition 9. —For g ≥ 1, we have 2g a B 2g g+1 λ λ = (−1) . g g−1 (2g)! 2g DR (a,−a) Proposition 9 is essentially equivalent to the basic calculation of [2, Theorem 6.5] in the Gromov-Witten theory of local curves. There are two minor differences. First, our answer differs from [2, Theorem 6.5] by a factor of 2g coming from the choice of branch point in [2, Theorem 6.5]. Second, if we write out the generating series obtained from Proposition 9 then we only obtain the cot(u) summand in the formula of [2,Theo- rem 6.5] after a standard correction accounting for the geometric difference between the space of rubber maps and the space of maps relative to three points. Proof. — We need only compute the contributions to DR (a, −a) of all rational tailstypegraphssince λ λ vanishes on all other strata. Since we just have two marked g g−1 points, there are exactly two rational tails type graphs: • the graph with a single vertex, • the graph with a vertex of genus g, a vertex of genus 0 containing both marked points, and exactly one edge joining the two vertices. The contribution of the ﬁrst graph is (using Lemma 8) 2g 2g a a B 1 2g g g+1 (ψ + ψ ) λ λ = · (−1) 1 2 g g−1 g g 2 g! 2 g! 2g (2g − 1)!! g,2 2g a B 2g g+1 = (−1) . (2g)! 2g The contribution of the second graph vanishes because the weight of the edge is equal to 0. For the above computation, we only require Hain’s formula for the restriction of the ct double ramiﬁcation cycle to M (the extension to M is not needed). In the forthcom- g,n g,n ing paper [22], we will study the Gromov-Witten theory of the rubber over the resolution of the A -singularity where the full structure is needed. n 258 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Acknowledgements We are grateful to A. Buryak, R. Cavalieri, A. Chiodo, E. Clader, C. Faber, G. Farkas, S. Grushevsky, J. Guéré, G. Oberdieck, D. Oprea, R. Vakil, and Q. Yin for discussions about double ramiﬁcation cycles and related topics. The Workshop on Pixton’s conjectures at ETH Zürich in October 2014 played an important role in our collaboration. A. P. and D. Z. have been frequent guests of the Forschungsinstitut für Mathematik (FIM) at ETHZ. The paper was completed at the conference Moduli spaces of holomorphic differen- tials at Humboldt University in Berlin in February 2016 attended by all four authors. F. J. was partially supported by the Swiss National Science Foundation grant SNF-200021143274. R. P. was partially supported by SNF-200021143274, SNF- 200020162928, ERC-2012-AdG-320368-MCSK, SwissMap, and the Einstein Stiftung. A. P. was supported by a fellowship from the Clay Mathematics Institute. D. Z. was supported by the grant ANR-09-JCJC-0104-01. Appendix A: Polynomiality by A. Pixton A.1 Overview Our goal here is to present a self contained proof of Proposition 3 of Section 1.4.Let A = (a ,..., a ) 1 n be a vector of k-twisted double ramiﬁcation data for genus g ≥ 0. A longer treatment, in d,k the context of the deeper polynomiality of P (A) in the parts a of A, will be given in [35]. Proposition 3 (Pixton [35]). — Let ∈ G be a ﬁxed stable graph. Let Q be a polynomial g,n with Q-coefﬁcients in variables corresponding to the set of half-edges H( ). Then the sum F(r) = Q w(h ), ...,w(h ) 1 |H( )| w∈W ,r,k h ( ) over all possible k-weightings mod r is (for all sufﬁciently large r) a polynomial in r divisible by r . A.2 Polynomiality of F(r) A.2.1 Ehrhart polynomials The polynomiality of F(r) for r sufﬁciently large will be derived as a consequence of a small variation of the standard theory [37] of Ehrhart polynomials of integral convex polytopes. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 259 Let M be a v ×hmatrix with Q-coefﬁcients satisfying the following two properties: (i) M is totally unimodular: every square minor of M has determinant 0, 1, or −1. (ii) If x = (x ) is a column vector satisfying j 1≤j≤h Mx =0and x ≥0forall j, then x = 0. We will use the notation x ≥ 0 to signify x ≥ 0for all j.Let h h Z = x | x ∈ Z and x ≥ 0 . ≥0 Proposition A1. —Let a, b ∈ Z be column vectors with integral coefﬁcients. Let Q ∈ Q[x ,..., x ] be a polynomial. Then, the sum over lattice points 1 h S(r) = Q(x) x∈Z , Mx=a+rb ≥0 is a polynomial in the integer parameter r (for all sufﬁciently large r). Proof. — We follow the treatment of Ehrhart theory in [37, Sections 4.5–4.6]. To start, we may assume all coefﬁcients a of a are nonnegative: if not, multiply the negative coefﬁcients of a and the corresponding parts of M and b by −1. Next, we deﬁne a generating function of h + v + 1 variables: R(X ,..., X , Y ,..., Y , Z) 1 h 1 v x y z = X Y Z . h v x∈Z , y∈Z , z∈Z , Mx=y+zb ≥0 ≥0 ≥0 By [37, Theorem 4.5.11], R represents a rational function of the variables X, Y, and Z with denominator given by a product of factors of the form 1 − Monomial(X, Y, Z). The monomials in the denominator factors correspond to the extremal rays of the cone h+v+1 in Z speciﬁed by the condition ≥0 Mx = y + zb. Because M is totally unimodular, these extremal rays all have integral generators with z-coordinate equal to 0 or 1. Therefore, the denominator is a product of factors of the form 1 − Monomial(X, Y) and 1 − Monomial(X, Y) · Z. The standard situation for the Ehrhart polynomial occurs when Q = 1 is the constant polynomial and a = 0. The variation required here is not very signiﬁcant, but we were not able to ﬁnd an adequate reference. 260 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE The sum S(r) in the statement of Proposition A1 can be obtained from R by exe- cuting the following steps: (i) apply a differential operator in the X variables to R which has the effect of x y z multiplying the coefﬁcient of X Y Z by Q(x), (ii) apply the differential operator a a a 1 2 v 1 ∂ ∂ ∂ ... , v a a 1 2 a ! ∂ Y ∂ Y ∂ Y 1 2 i=1 v (iii) set X = 1for all1 ≤ j ≤ hand Y = 0for all1 ≤ i ≤ v, j i (iv) extract the coefﬁcient of Z . After step (iii) and before step (iv), we have a rational function in Z with denominator equal to a power of 1 − Z. The Z coefﬁcient of such a rational function is (eventually) a polynomial in r. A.2.2 Vertex-edge matrix We now prove the polynomiality of F(r) for all r sufﬁciently large. Let A = (a ,..., a ) 1 n be a vector of k-twisted double ramiﬁcation data for genus g ≥ 0. Let ∈ G be a ﬁxed g,n stable graph. For all sufﬁciently large r, the sum over half-edge weightings (31) w ∈ W , 0 ≤ w(h)< r ,r,k satisfying conditions mod r, (32)F(r) = Q w(h ), ...,w(h ) , 1 |H( )| w∈W ,r,k can be split into ﬁnitely many sums over half-edge weightings satisfying equalities (in- volving multiples of r). We study each sum in the splitting of (32) separately. The conditions on the weights w(h) in these sums will be of the following form. The inequalities 0 ≤ w(h)< r continue to hold. Moreover, each leg, edge, and vertex yields an equality: By property (ii) of the deﬁnition of M, the specialization is well deﬁned. There are no denominator factors in R of the form 1 − Monomial(X). Equalities not equalities mod r . DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 261 (i) w(h ) = a + rb at the ith leg h ,where i i i i b =0if a ≥0and b =1if a < 0, i i i i (ii) w(h) + w(h ) = 0or w(h) + w(h ) = r along the edge (h, h ), (iii) w(h) = k(2g(v) − 2 + n(v)) + rb at the vertex v with b ∈ Z. v v v(h)=v The conditions w(h )< r are now redundant with the equalities of type (i) and can be removed. Along an edge (h, h ), the condition w(h)< r only serves to rule out the possibility w(h) = r, w(h ) = 0 (if we have the equality w(h) + w(h ) = r). Thus the condition w(h)< r can be removed if we subtract off the contributions with w(h) = r, w(h ) = 0. These contributions are given by more sums of the same form (corresponding to the graph formed by deleting the edge (h, h ) from ), multiplied by powers of r and 0 coming from factors of w(h) and w(h ) in the polynomial Q. Thus the original sum F(r) can be split into a linear combination (with coefﬁcients being polynomials in r) of sums with weight conditions of the following form: inequalities w(h) ≥ 0 and equalities of the form (i), (ii), and (iii) associated to legs, edges, and vertices as de- scribed above. We must check that the resulting sums satisfy the hypotheses of Proposition A1. Consider the graph obtained from by adding a new vertex in the middle of each edge e ∈ E( ) and at the end of each leg l ∈ L( ). The half-edge set of then corresponds bijectively to the edge set of , H( ) ←→ E . We associate each linear condition on the half-edge weights w(h) described in (i)–(iii) above to the corresponding vertex of . For types (i) and (ii), the corresponding vertex of is a new vertex. For (iii), the corresponding vertex already exists in . The matrix M associated to the linear terms in these linear conditions is precisely the vertex-edge incidence matrix of :Mis av × hmatrix, v = V , h = E = H( ) , with matrix entry 1 if the corresponding vertex of meets the corresponding edge of (and 0 otherwise). Since all the entries of M are nonnegative and every column contains a positive entry, we immediately see (Mx = 0and x ≥ 0) ⇒ x = 0. Because is bipartite, the incidence matrix M is totally unimodular by Proposition A2 below. We may therefore apply Proposition A1 to each sum of the splitting of (32). Each such sum is a polynomial in r for sufﬁciently large r.Hence, F(r) is a polynomial in r for sufﬁciently large r. 262 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE Proposition A2. — The vertex-edge matrix of a bipartite graph is totally unimodular. A proof of general results concerning the total unimodularity of vertex-edge ma- trices can be found in [20]. Proposition A2 arises as a special case. h ( ) A.3 Divisibility by r In this section we write b = h ( ) for convenience. We wish to show that the polynomial F(r) is divisible by r . This is implied by checking that the valuation condition, val F(p) ≥ b, holds for all sufﬁciently large primes p.Here, val : Q → Z is the p-adic valuation. Let h =|H( )| be the number of half-edges of . We write the evaluation F(p) as F(p) = Q(w ,...,w ) 1 h 0≤w ,...,w ≤p−1 1 h where the sum is over h-tuples (w ,...,w ) subject to h − b linear conditions mod p. 1 h The polynomial Q has (p-integral) Q-coefﬁcients. We count the linear conditions mod p as follows. By the deﬁnition of a k-weighting mod p, we start with v = L( ) + E( ) + V( ) linear conditions, but the conditions are not independent since the vector A of k-twisted double ramiﬁcation data satisﬁes a = k(2g − 2 + n). i=1 After removing a redundant linear relation, we have v − 1 independent linear conditions. Since h = H( ) = L( ) + 2 E( ) , and is connected, we see h − b = h − E( ) − V( ) + 1 = v − 1. DOUBLE RAMIFICATION CYCLES ON THE MODULI SPACES OF CURVES 263 Let Q be a polynomial with Q-coefﬁcients in h variables, with all coefﬁcients p-integral. We will prove every summation (33) F(p) = Q(w ,...,w ), 1 h 0≤w ,...,w ≤p−1 1 h where the sum is over h-tuples (w ,...,w ) subject to h − b linear conditions mod p,is 1 h divisible by p for p ≥ deg(Q) + 2. We maycertainlytake Q to be a monomial. Without loss of generality, we need only consider Q(w ,...,w ) = w . 1 h j j=1 We write the linear conditions imposed in the sum (33)as m w + =0mod p ij j i j=1 for 1 ≤ i ≤ h − b. Alternatively, we may impose the conditions by standard p-th root of unity ﬁlters: m w h−b ij j p F(p) = w · z · z . i i z ,...,z 0≤w ,...,w ≤p−1 j=1 1≤i≤h−b 1≤i≤h−b 1 h−b 1 h 1≤j≤h The ﬁrst sum on the right is over all (h − b)-tuples (z ,..., z ) of p-th roots of unity. 1 h−b The second sum on the right is over all h-tuples (w ,...,w ). 1 h Consider the following polynomial with integer coefﬁcients: p p+1 Z − pZ + (p − 1)Z G(Z) = aZ = . (1 − Z) 0≤a≤p−1 We can use G to rewrite the terms inside the ﬁrst sum in the above expression: ij h−b (34) p F(p) = G z · z . i i z ,...,z 1≤j≤h 1≤i≤h−b 1≤i≤h−b 1 h−b 24 2 To increase w to w , add a new variable w and a new linear condition w = w . If a variable w does not 1 h+1 1 h+1 j appear in Q, then a reduction to a fewer variable case can be made. 264 F. JANDA, R. PANDHARIPANDE, A. PIXTON, D. ZVONKINE p−2 For every p-th root of unity z,G(z) has p-adic valuation at least : either z = 1 p−1 p(p−1) and G(z) = or p 1 z = 1 ⇒ G(z) = and val (z − 1) = . z − 1 p − 1 Therefore, every term in the sum on the right side of (34)has p-adic valuation at least p−2 h · .Hence, p−1 p − 2 val F(p) ≥ h · − (h − b). p − 1 Since val (F(p)) is an integer, val F(p) ≥ b if p ≥ h + 2. d,k A.4 Polynomiality of P (A) in the a For double ramiﬁcation data of ﬁxed length n,consider d,k d P (a ,..., a ) ∈ R (M ) 1 n g,n as a function of (a ,..., a ) ∈ Z , a = k(2g − 2 + n) 1 n i i=1 holding g, d,and k ﬁxed. The following property is proven in [35]: d,k P (a ,..., a ) is polynomial in the variables a . 1 n i By Theorem 1, the double ramiﬁcation cycle DR (A) is then also polynomial in the variables a . Shadows of the polynomiality of the double ramiﬁcation cycle were proven earlier in [3, 29]. 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PIXTON, On combinatorial properties of the explicit expression for double ramiﬁcation cycles, in preparation. 36. B. RIEMANN, Theorie der Abel’schen Functionen, J. Reine Angew. Math., 54 (1857), 115–155. 37. R. STANLEY, Enumerative Combinatorics: Vol I, Cambridge University Press, Cambridge, 1999. 38. G. van der GEER, Cycles on the moduli space of abelian varieties, in Moduli of Curves and Abelian Varieties, Aspects Math., vol. E33, pp. 65–89, Vieweg, Braunschweig, 1999. F. J. Department of Mathematics, University of Michigan, Ann Arbor, USA janda@umich.edu R. P. Departement Mathematik, ETH Zürich, Zürich, Switzerland rahul@math.ethz.ch A. P. Department of Mathematics, MIT, Cambridge, USA apixton@mit.edu D. Z. Institut de Mathématiques de Jussieu, CNRS, Paris, France dimitri.zvonkine@imj-prg.fr Manuscrit reçu le 27 mars 2016 Manuscrit accepté le 20 mars 2017 publié en ligne le 10 mai 2017.

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