Relations in the tautological ring of the moduli space of curves

Transcription

1 Relations in the tautological ring of the moduli space of curves R. Pandharipande and A. Pixton January 2013 Abstract The virtual geometry of the moduli space of stable quotients is used to obtain Chow relations among the κ classes on the moduli space of nonsingular genus g curves. In a series of steps, the stable quotient relations are rewritten in successively simpler forms. The final result is the proof of the Faber-Zagier relations conjectured in Contents 0 Introduction 2 1 Classical vanishing relations 8 2 Stable quotients 16 3 Stable quotients relations 19 4 Analysis of the relations 31 5 Transformation 38 6 Equivalence 46 1

2 0 Introduction 0.1 Tautological classes For g 2, let M g be the moduli space of nonsingular, projective, genus g curves over C, and let π : C g M g 1 be the universal curve. We view M g and C g as nonsingular, quasi-projective, Deligne-Mumford stacks. However, the orbifold perspective is sufficient for most of our purposes. The relative dualizing sheaf ω π of the morphism 1 is used to define the cotangent line class ψ = c 1 ω π A 1 C g, Q. The κ classes are defined by push-forward, The tautological ring κ r = π ψ r+1 A r M g. R M g A M g, Q is the Q-subalgebra generated by all of the κ classes. Since κ 0 = 2g 2 Q is a multiple of the fundamental class, we need not take κ 0 as a generator. There is a canonical quotient Qκ 1, κ 2, κ 3,...] q R M g 0. We study here the ideal of relations among the κ classes, the kernel of q. We may also define a tautological ring RH M g H M g, Q generated by the κ classes in cohomology. Since there is a natural factoring Qκ 1, κ 2, κ 3,...] q R c M g RH M g via the cycle class map c, algebraic relations among the κ classes are also cohomological relations. Whether or not there exist more cohomological relations is not yet settled. 2

3 There are two basic motivations for the study of the tautological rings R M g. The first is Mumford s conjecture, proven in 2002 by Madsen and Weiss 11], lim g H M g, Q = Qκ 1, κ 2, κ 3,...], determining the stable cohomology of the moduli of curves. While the κ classes do not exhaust H M g, Q, there are no other stable classes. The study of R M g undertaken here is from the opposite perspective we are interested in the ring of κ classes for fixed g. The second motivation is from a large body of cycle class calculations on M g often related to Brill-Noether theory. The answers invariably lie in the tautological ring R M g. The first definition of the tautological rings by Mumford 14] was at least partially motivated by such algebro-geometric cycle constructions. 0.2 Faber-Zagier conjecture Faber and Zagier have conjectured a remarkable set of relations among the κ classes in R M g. Our main result is a proof of the Faber-Zagier relations, stated as Theorem 1 below, by a geometric construction involving the virtual class of the moduli space of stable quotients. To write the Faber-Zagier relations, we will require the following notation. Let the variable set p = { p 1, p 3, p 4, p 6, p 7, p 9, p 10,... } be indexed by positive integers not congruent to 2 modulo 3. series Ψt, p = 1 + tp 3 + t 2 p 6 + t 3 p i=0 6i! 3i!2i! ti + p 1 + tp 4 + t 2 p i=0 Define the 6i! 6i + 1 3i!2i! 6i 1 ti. Since Ψ has constant term 1, we may take the logarithm. Define the constants Cr FZ σ by the formula logψ = σ r=0 3 C FZ r σ t r p σ.

4 The above sum is over all partitions σ of size σ which avoid parts congruent to 2 modulo 3. The empty partition is included in the sum. To the partition σ = 1 n 1 3 n 3 4 n4, we associate the monomial p σ = p n 1 1 p n 3 3 p n 4 4. Let γ FZ = Cr FZ σ κ r t r p σ. σ r=0 For a series Θ Qκ]t, p]] in the variables κ i, t, and p j, let Θ] t r p σ denote the coefficient of t r p σ which is a polynomial in the κ i. Theorem 1. In R r M g, the Faber-Zagier relation exp γ FZ ] t r p σ = 0 holds when g 1 + σ < 3r and g r + σ + 1 mod 2. The dependence upon the genus g in the Faber-Zagier relations of Theorem 1 occurs in the inequality, the modulo 2 restriction, and via κ 0 = 2g 2. For a given genus g and codimension r, Theorem 1 provides only finitely many relations. While not immediately clear from the definition, the Q-linear span of the Faber-Zagier relations determines an ideal in Qκ 1, κ 2, κ 3,...] the matter is discussed in Section 6 and a subset of the Faber-Zagier relations generating the same ideal is described. As a corollary of our proof of Theorem 1 via the moduli space of stable quotients, we obtain the following stronger boundary result. If g 1+ σ < 3r and g r + σ + 1 mod 2, then exp γ FZ ] t r p σ R M g. 2 Not only is the Faber-Zagier relation 0 on R M g, but the relation is equal to a tautological class on the boundary of the moduli space M g. A precise conjecture for the boundary terms has been proposed in 18]. 0.3 Gorenstein rings By results of Faber 3] and Looijenga 10], we have dim Q R g 2 M g = 1, R >g 2 M g = 0. 3 A canonical parameterization of R g 2 M g is obtained via integration. Let E M g 4

5 be the Hodge bundle with fiber H 0 C, ω C over the moduli point C] M g. Let λ k denote the k th Chern class of E. The linear map ɛ ɛ : Qκ 1, κ 2, κ 3,...] Q, fκ fκ λ g λ g 1 M g factors through R M g and determines an isomorphism via the non-trivial evaluation M g κ g 2 λ g λ g 1 = ɛ : R g 2 M g = Q 1 B 2g 2 2g 1 2g 1!! 2g. 4 A survey of the construction and properties of ɛ can be found in 5]. The evaluations under ɛ of all polynomials in the κ classes are determined by the following formulas. First, the Virasoro constraints for surfaces 7] imply a related evaluation previously conjectured in 3]: M g,n ψ α 1 1 ψ αn n λ g λ g 1 = 2g + n 3!2g 1!! 2g 1! n i=1 2α i 1!! where α i > 0. Second, a basic relation due to Faber holds: M g κ g 2 λ g λ g 1, 5 M g,n ψ α 1 1 ψ αn n λ g λ g 1 = σ S n M g κ σ λ g λ g 1. 6 The sum on the right is over all elements of the symmetric group S n, κ σ = κ c1... κ cr where c 1,..., c r is the set partition obtained from the cycle decomposition of σ, and c i = j c i α j 1. Relation 6 is triangular and can be inverted to express the ɛ evaluations of the κ monomials in terms of 5. Computations of the tautological rings in low genera led Faber to formulate the following conjecture in

6 Conjecture 1. For all g 2 and all 0 k g 2, the pairing is perfect. R k M g R g 2 k M g ɛ Q 7 The pairing 7 is the ring multiplication of R M g composed with ɛ. A perfect pairing identifies the first vector space with the dual of the second. If Faber s conjecture is true in genus g, then R M g is a Gorenstein local ring. Let I g R M g be the ideal determined by the kernel of the pairing 7 in Faber s conjecture. Define the Gorenstein quotient R GM g = R M g I g. If Faber s conjecture is true for g, then I g = 0 and RG M g = R M g. The pairing 7 can be evaluated directly on polynomials in the κ classes via 4-6. The Gorenstein quotient RG M g is completely determined by the κ evaluations and the ranks 3. The ring RG M g can therefore be studied as a purely algebro-combinatorial object. Faber and Zagier conjectured the relations of Theorem 1 from a concentrated study of the Gorenstein quotient RG M g. The Faber-Zagier relations were first written in 2000 and were proven to hold in RG M g in The validity of the Faber-Zagier relations in R M g has been an open question since then. 0.4 Other relations? By substantial computation, Faber has verified Conjecture 1 holds for genus g < 24. Moreover, his calculations show the Faber-Zagier set yields all relations among κ classes in R M g for g < 24. However, he finds the Faber-Zagier relations of Theorem 1 do not yield a Gorenstein quotient in genus 24. Let FZ g Qκ 1, κ 2, κ 3,...] be the ideal determined by the Faber-Zagier relations of Theorem 1, and let R FZM g = Qκ 1, κ 2, κ 3,...] FZ g. 6

7 Faber finds a mismatch in codimension 12, R 12 FZM 24 R 12 G M Exactly 1 more relation holds in the Gorenstein quotient. To the best of our knowledge, a relation in R M g which is not in the span of the Faber-Zagier relations of Theorem 1 has not yet been found. The following prediction is consistent with all present calculations. Conjecture 2. For all g 2, the kernel of Qκ 1, κ 2, κ 3,...] is the Faber-Zagier ideal FZ g. q R M g 0 Conjectures 1 and 2 are both true for g < 24. By the inequality 8, Conjectures 1 and 2 can not both be true for all g. Which is false? Finally, we note the above discussion might have a different outcome if the tautological ring RH M g in cohomology is considered instead. Perhaps there are more relations in cohomology? These questions provide a very interesting line of inquiry. 0.5 Plan of the paper We start the paper in Section 1 with a modern treatment of Faber s classical construction of relations among the κ classes. The result, in Wick form, is stated as Theorem 2 of Section 1.2. While the outcome is an effective source of relations, their complexity has so far defied a complete analysis. After reviewing stable quotients on curves in Section 2, we derive an explicit set of κ relations from the virtual geometry of the moduli space of stable quotients in Section 3. The resulting equations are more tractable than those obtained by classical methods. In a series of steps, the stable quotient relations are transformed to simpler and simpler forms. The first step, Theorem 5, comes almost immediately from the virtual localization formula 8] applied to the moduli space of stable quotients. After further analysis in Section 4, the simpler form of Proposition 10 is found. A change of variables is applied in Section 5 that transforms the relations to Proposition 15. Our final result, Theorem 1, establishes the previously conjectural set of tautological relations proposed more than a decade ago by Faber and Zagier. The proof of Theorem 1 is completed in Section 6. 7

8 A natural question is whether Theorem 1 can be extended to yield explicit relations in the tautological ring of M g,n. A precise conjecture of exactly such an extension is given in 18]. There is no doubt that our methods here can also be applied to investigate tautological relations in M g,n. Whether the simple form of 18] will be obtained remains to be seen. A different method, valid only in cohomology, of approaching the conjecture of 18] is pursued in 17]. 0.6 Acknowledgements We first presented our proof of the Faber-Zagier relations in a series of lectures at Humboldt University in Berlin during the conference Intersection theory on moduli space in A detailed set of notes, which is the origin of the current paper, is available 16]. We thank G. Farkas for the invitation to speak there. Discussions with C. Faber played an important role in the development of our ideas. The research reported here was done during visits of A.P. to IST Lisbon during the year The paper was written at ETH Zürich during the year R.P. was supported in Lisbon by a Marie Curie fellowship and a grant from the Gulbenkian foundation. In Zürich, R.P. was partially supported by the Swiss National Science Foundation grant SNF A.P. was supported by a NDSEG graduate fellowship. 1 Classical vanishing relations 1.1 Construction Faber s original relations in his article Conjectural description of the tautological ring 3] are obtained from a very simple geometric construction. As before, let π : C g M g be the universal curve over the moduli space, and let π d : C d g M g be the map associated to the d th fiber product of the universal curve. For every point C, p 1,..., p d ] C d g, we have the restriction map H 0 C, ω C H 0 C, ω C p p d. 9 8

9 Since the canonical bundle ω C has degree 2g 2, the restriction map is injective if d > 2g 2. Let Ω d C d g be the rank d bundle with fiber H 0 C, ω C p p d over the moduli point C, p 1,..., p d ] C d g. If d > 2g 2, the restriction map 9 yields an exact sequence over C d, 0 E Ω d Q d g 0 where E is the rank g Hodge bundle and Q d g is the quotient bundle of rank d g. We see c k Q d g = 0 A k C d g for k > d g. After cutting the vanishing Chern classes c k Q d g down with cotangent line and diagonal classes in C d g and pushing-forward via π d to M g, we arrive at Faber s relations in R M g. 1.2 Wick form From our point of view, at the center of Faber s relations in 3] is the function Θt, x = d d=0 i=1 1 + it 1d d! x d t d. The differential equation tx + 1 d Θ + t + 1Θ = 0 dx is easily found. Hence, we obtain the following result. Lemma 1. Θ = 1 + x t+1 t. We introduce a variable set z indexed by pairs of integers z = { z i,j i 1, j i 1 }. For monomials z σ = i,j z σ i,j i,j, 9

10 we define lσ = i,j iσ i,j, σ = i,j jσ i,j. Of course Autσ = i,j σ i,j!. The variables z are used to define a differential operator D = z i,j t j x d i. dx i,j After applying expd to Θ, we obtain Θ D = expd Θ = d σ d=0 i=1 1 + it 1d d! x d t d d lσ t σ z σ Autσ where σ runs over all monomials in the variables z. Define constants C d r σ by the formula logθ D = σ d=1 r= 1 C d r σ t r xd d! zσ. By an elementary application of Wick s formula as explained in Section below, the t dependence of logθ D has at most simple poles. Finally, we consider the following function, γ F = i 1 B 2i 2i2i 1 κ 2i 1t 2i 1 + σ d=1 r= 1 C d r σ κ r t r xd d! zσ. 10 The Bernoulli numbers appear in the first term, k=0 B k u k k! = u e u 1. Denote the t r x d z σ coefficient of exp γ F by exp γf ] t r x d z σ Qκ 1, κ 0, κ 1, κ 2,...]. Our form of Faber s equations is the following result. 10

11 Theorem 2. In R r M g, the relation exp γf ] t r x d z σ = 0 holds when r > g + σ and d > 2g 2. In the tautological ring R M g, the standard conventions κ 1 = 0, κ 0 = 2g 2 are followed. For fixed g and r, Theorem 2 provides infinitely many relations by increasing d. The variables z i,j efficiently encode both the cotangent and diagonal operations studied in 3]. In particular, the relations of Theorem 2 are equivalent to a mixing of all cotangent and diagonal operations studied there. The proof of Theorem 2 is presented in Section 1.3. While Theorem 2 has an appealingly simple geometric origin, the relations do not seem to fit the other forms we will see later. In particular, we do not know how to derive Theorem 1 from Theorem 2. Extensive computer calculations by Faber suggest the following. Conjecture 3. For all g 2, the relations of Theorem 2 are equivalent to the Faber-Zagier relations. In particular, despite significant effort, the relation in RG 12M 24 which is missing in RFZ 12 M 24 has not been found via Theorem 2. Other geometric strategies have so far also failed to find the missing relation 19, 20]. 1.3 Proof of Theorem The Chern roots of Ω d Let ψ i A 1 C d g, Q be the first Chern class of the relative dualizing sheaf ω π pulled back from the i th factor, C d g C g. For i j, let D ij A 1 C d g, Q be the class of the diagonal C g C 2 g pulled-back from the product of the i th and j th factors, C d g C 2 g. 11

12 Finally, let i = D 1,i D i 1,i A 1 C d g, Q following the convention 1 = 0. The Chern roots of Ω d, c t Ω d = d 1 + ψ i i t 11 i=1 = 1 + ψ 1 t 1 + ψ 2 D 12 t d ψ d D id t are obtained by a simple induction, see 3]. We may expand the right side of 11 fully. The resulting expression is a polynomial in the d + d 2 variables. ψ 1,..., ψ d, D 12, D 13,..., D d 1,d. The sign on the diagonal variables is chosen because of the self-intersection formula D ij 2 = ψ i D ij = ψ j D ij. Let M d r denote the coefficient in degree r, i=1 c t Ω d = Mr d ψ i, D ij t r. r=0 Lemma 2. After setting all the variables to 1, Mr d ψ i = 1, D ij = 1 t r = r=0 d 1 + it. i=1 Proof. The results follows immediately from the Chern roots 11. Lemma 2 may be viewed counting the number of terms in the expansion of the total Chern class c t Ω d. 12

13 1.3.2 Connected counts A monomial in the diagonal variables D 12, D 13,..., D d 1,d 12 determines a set partition of {1,..., d} by the diagonal associations. example, the monomial 3D12D 2 1,3 D56 3 determines the set partition For {1, 2, 3} {4} {5, 6} in the d = 6 case. A monomial in the variables 12 is connected if the corresponding set partition consists of a single part with d elements. A monomial in the variables ψ 1,..., ψ d, D 12, D 13,..., D d 1,d 13 is connected if the corresponding monomial in the diagonal variables obtained by setting all ψ i = 1 is connected. Let S d r be the summand of the evaluation M d r ψ i = 1, D ij = 1 consisting of the contributions of only the connected monomials. Lemma 3. We have d d=1 r=0 d=1 r=0 Sr d t r xd d! = log 1 + d d=1 i=1 1 + it xd d! Proof. By a standard application of Wick s formula, the connected and disconnected counts are related by exponentiation, d exp Sr d t r xd = 1 + Mr d ψ i = 1, D ij = 1 t r xd d! d!. d=1 r=0 The right side is then evaluated by Lemma 2. Since a connected monomial in the variables 13 must have at least d 1 factors of the variables D ij, we see Sr d = 0 if r < d 1. Using the selfintersection formulas, we obtain d π d cr Ω d t r xd d! = exp d Sr d 1 d 1 κ r d t r xd. 14 d! d=1 r=0 d=1 r=0 13.

14 To account for the alternating factor 1 d 1 and the κ subscript, we define the coefficients C d r by d d=1 r 1 Cr d t r xd d! = log 1 + d d=1 i=1 1 + it 1d t d x d d! The vanishing S d r<d 1 = 0 implies the vanishing Cd r< 1 = 0. The formula for the total Chern class of the Hodge bundle E on M g follows immediately from Mumford s Grothendieck-Riemann-Roch calculation 14], c t E = i 1 B 2i 2i2i 1 κ 2i 1t 2i 1. Putting the above results together yields the following formula: π d cr Q d g r d xd t d=1 r Cutting exp i 1 d! = B 2i 2i2i 1 κ 2i 1t 2i 1 For d > 2g 2 and r > d g, we have the vanishing c r Q d g = 0 A r C d g, Q. d=1 r 1. Cr d κ r t r xd d! Before pushing-forward via π d, we will cut c r Q d g with products of classes in A C d g, Q. With the correct choice of cutting classes, we will obtain the relations of Theorem 2. Let a, b be a pair of integers satisfying a 0 and b 1. We define the cutting class φa, b] = 1 b 1 I =b ψ a I D I 15. where I {1,..., d} is subset of order b, D I A b 1 C d g, Q is the class of the corresponding small diagonal, and ψ I is the cotangent line at the point indexed by I. The class ψ I is well-defined on the small diagonal indexed by 14

15 I. The degree of φa, b] is a + b 1. The number of terms on the right side of 15 is a degree b polynomial in d, d = db b b! d 1b 1 b with no constant term. The sign 1 b 1 in definition 15 is chosen to match the sign conventions of the Wick analysis in Section For example, φ0, 2] = D ij, φ0, 3] = D ij D jk. i<j i<j<k The number of terms means the evaluation at ψ I = 1 and D ij = 1. A better choice of cutting class is obtained by the following observation. For every pair of integers i, j with i 1 and j i 1, we can find a unique linear combination Φi, j] = λ a,b φa, b], λ a,b Q a+b 1=j for which the evaluation of Φi, j] at ψ I = 1 and D ij = 1 is d i. By definition, Φi, j] is of pure degree j Full Wick form We repeat the Wick analysis of Section for the Chern class of Q d g cut by the classes Φi, j] in order to write a formula for π d exp z i,j t j Φi, j] c r Q d g t r 1 x d t d d! d=1 r 0 i,j where the sum in the argument of the exponential is over all i 1 and j i 1. The variable set z introduced in Section 1.2 appears here. Since Φi, j] yields d i after evaluation at ψ I = 1 and D ij = 1 and is of pure degree j, we conclude π d exp z i,j t j Φi, j] c r Q d g t r 1 x d t d d! = exp γf. 16 d=1 r 0 i,j 15

16 Let d > 2g 2. Since c s Q d g = 0 for s > d g, the t r x d z σ coefficient of 16 vanishes if r + d σ > d g which is equivalent to r > g + σ. The proof of Theorem 2 is complete. 2 Stable quotients 2.1 Stability Our proof of the Faber-Zagier relations in R M g will be obtained from the virtual geometry of the moduli space of stable quotients. We start by reviewing the basic definitions and results of 13]. Let C be a curve which is reduced and connected and has at worst nodal singularities. We require here only unpointed curves. See 13] for the definitions in the pointed case. Let q be a quotient of the rank N trivial bundle C, C N q O C Q 0. If the quotient subsheaf Q is locally free at the nodes and markings of C, then q is a quasi-stable quotient. Quasi-stability of q implies the associated kernel, 0 S C N q O C Q 0, is a locally free sheaf on C. Let r denote the rank of S. Let C be a curve equipped with a quasi-stable quotient q. The data C, q determine a stable quotient if the Q-line bundle ω C r S ɛ 17 is ample on C for every strictly positive ɛ Q. Quotient stability implies 2g 2 0. Viewed in concrete terms, no amount of positivity of S can stabilize a genus 0 component P 1 = P C unless P contains at least 2 nodes or markings. If P contains exactly 2 nodes or markings, then S must have positive degree. A stable quotient C, q yields a rational map from the underlying curve C to the Grassmannian Gr, N. We will only require the G1, 2 = P 1 case for the proof Theorem 1. 16

17 2.2 Isomorphism Let C be a curve. Two quasi-stable quotients C N O C q Q 0, C N q O C Q 0 18 on C are strongly isomorphic if the associated kernels S, S C N O C are equal. An isomorphism of quasi-stable quotients is an isomorphism of curves φ : C, q C, q φ : C C such that the quotients q and φ q are strongly isomorphic. Quasi-stable quotients 18 on the same curve C may be isomorphic without being strongly isomorphic. The following result is proven in 13] by Quot scheme methods from the perspective of geometry relative to a divisor. Theorem 3. The moduli space of stable quotients Q g Gr, N, d parameterizing the data C, 0 S C N q O C Q 0, with ranks = r and degs = d, is a separated and proper Deligne- Mumford stack of finite type over C. 2.3 Structures Over the moduli space of stable quotients, there is a universal curve with a universal quotient π : U Q g Gr, N, d 19 0 S U C N O U q U Q U 0. The subsheaf S U is locally free on U because of the stability condition. 17

18 The moduli space Q g Gr, N, d is equipped with two basic types of maps. If 2g 2 > 0, then the stabilization of C determines a map ν : Q g Gr, N, d M g by forgetting the quotient. The general linear group GL N C acts on Q g Gr, N, d via the standard action on C N O C. The structures π, q U, ν and the evaluations maps are all GL N C-equivariant. 2.4 Obstruction theory The moduli of stable quotients maps to the Artin stack of pointed domain curves ν A : Q g Gr, N, d M g. The moduli of stable quotients with fixed underlying curve C] M g is simply an open set of the Quot scheme of C. The following result of 13, Section 3.2] is obtained from the standard deformation theory of the Quot scheme. Theorem 4. The deformation theory of the Quot scheme determines a 2- term obstruction theory on the moduli space Q g Gr, N, d relative to ν A given by RHomS, Q. More concretely, for the stable quotient, 0 S C N O C q Q 0, the deformation and obstruction spaces relative to ν A are HomS, Q and Ext 1 S, Q respectively. Since S is locally free, the higher obstructions Ext k S, Q = H k C, S Q = 0, k > 1 vanish since C is a curve. An absolute 2-term obstruction theory on the moduli space Q g Gr, N, d is obtained from Theorem 4 and the smoothness of M g, see 1, 2, 7]. The analogue of Theorem 4 for the Quot scheme of a fixed nonsingular curve was observed in 12]. The GL N C-action lifts to the obstruction theory, and the resulting virtual class is defined in GL N C-equivariant cycle theory, Q g Gr, N, d] vir A GL N C Q g Gr, N, d. 18

19 For the construction of the Faber-Zagier relation, we are mainly interested in the open stable quotient space ν : Q g P 1, d M g which is simply the corresponding relative Hilbert scheme. However, we will require the full stable quotient space Q g P 1, d to prove the Faber-Zagier relations can be completed over M g with tautological boundary terms. 3 Stable quotients relations 3.1 First statement Our relations in the tautological ring R M g obtained from the moduli of stable quotients are based on the function Φt, x = d d=0 i=1 1 1 it 1 d d! x d t d. 20 Define the coefficients C d r by the logarithm, logφ = d=1 r= 1 C d r t r xd d!. Again, by an application of Wick s formula in Section 3.3, the t dependence has at most a simple pole. Let γ = i 1 B 2i 2i2i 1 κ 2i 1t 2i 1 + d=1 r= 1 C d r κ r t r xd d!. 21 Denote the t r x d coefficient of exp γ by exp γ ] t r x d Qκ 1, κ 0, κ 1, κ 2,...]. In fact, exp γ] t r x is homogeneous of degree r in the κ classes. d The first form of the tautological relations obtained from the moduli of stable quotients is given by the following result. 19

20 Proposition 4. In R r M g, the relation exp γ ] t r x d = 0 holds when g 2d 1 < r and g r + 1 mod 2. For fixed r and d, if Proposition 4 applies in genus g, then Proposition 4 applies in genera h = g 2δ for all natural numbers δ N. The genus shifting mod 2 property is present also in the Faber-Zagier relations. 3.2 K-theory class F d For genus g 2, we consider as before π d : C d g M g, the d-fold product of the universal curve over M g. Given an element C, p 1,..., p d ] C d g, there is a canonically associated stable quotient 0 O C Consider the universal curve d p j O C Q j=1 ɛ : U C d g with universal quotient sequence 0 S U O U Q U 0 obtained from 22. Let F d = Rɛ SU KC d g be the class in K-theory. For example, F 0 = E C is the dual of the Hodge bundle minus a rank 1 trivial bundle. 20

21 By Riemann-Roch, the rank of F d is r g d = g d 1. However, F d is not always represented by a bundle. By the derivation of 13, Section 4.6], F d = E B d C, 23 where B d has fiber H 0 C, O C d j=1 p j d j=1 p j over C, p 1,..., p d ]. The Chern classes of F d can be easily computed. Recall the divisor D i,j where the markings p i and p j coincide. Set i = D 1,i D i 1,i, with the convention 1 = 0. Over C, p 1,..., p d ], the virtual bundle F d is the formal difference H 1 O C p p d H 0 O C p p d. Taking the cohomology of the exact sequence 0 O C p p d 1 O C p p d O C p p d pd 0, we find Inductively, we obtain cf d = cf d d ψ d. cf d = Equivalently, we have c B d = 3.3 Proof of Proposition 4 Consider the proper morphism ce ψ d ψ d ψ d ψ d. 24 ν : Q g P 1, d M g. 21

22 Certainly the class ν 0 c Q g P 1, d] vir A M g, Q, 25 where 0 is the first Chern class of the trivial bundle, vanishes if c > 0. Proposition 4 is proven by calculating 25 by localization. We will find Proposition 4 is a subset of the much richer family of relations of Theorem 5 of Section 3.4. Let the torus C act on a 2-dimensional vector space V = C 2 with diagonal weights 0, 1]. The C -action lifts canonically to PV and Q g PV, d. We lift the C -action to a rank 1 trivial bundle on Q g PV, d by specifying fiber weight 1. The choices determine a C -lift of the class 0 c Q g PV, d] vir A 2d+2g 2 c Q g PV, d, Q. The push-forward 25 is determined by the virtual localization formula 7]. There are only two C -fixed loci. The first corresponds to a vertex lying over 0 PV. The locus is isomorphic to C d g / S d and the associated subsheaf 22 lies in the first factor of V O C when considered as a stable quotient in the moduli space Q g PV, d. Similarly, the second fixed locus corresponds to a vertex lying over PV. The localization contribution of the first locus to 25 is 1 d! πd c g d 1+c F d where π d : C d g M g. Let c F d denote the total Chern class of F d evaluated at 1. The localization contribution of the second locus is 1 g d 1 ] g d 1+c π d c F d d! where γ] k is the part of γ in A k C d g, Q. Both localization contributions are found by straightforward expansion of the vertex formulas of 13, Section 7.4.2]. Summing the contributions yields π d ] g d 1+c c g d 1+c F d + 1 g d 1 c F d for c > 0. We obtain the following result. 22 = 0 in R M g

23 Lemma 5. For c > 0 and c 0 mod 2, c g d 1+c F d = 0 in R M g. π d For c > 0, the relation of Lemma 5 lies in R r M g where r = g 2d 1 + c. Moreover, the relation is trivial unless g d 1 g d 1 + c = r d mod We may expand the right side of 24 fully. The resulting expression is a polynomial in the d + d 2 variables. ψ 1,..., ψ d, D 12, D 13,..., D d 1,d. Let M d r denote the coefficient in degree r, c t B d = M r d ψ i, D ij t r. r=0 Let S d r be the summand of the evaluation M d r ψ i = 1, D ij = 1 consisting of the contributions of only the connected monomials. Lemma 6. We have d=1 r=0 S r d t r xd d! = log 1 + d d=1 i=1 1 1 it x d d!. Proof. As before, by Wick s formula, the connected and disconnected counts are related by exponentiation, exp S r d t r xd = 1 + M r d d! ψ i = 1, D ij = 1 t r xd d!. d=1 r=0 d=1 r=0 23

24 Since a connected monomial in the variables ψ i and D ij must have at least d 1 factors of the variables D ij, we see S r d = 0 if r < d 1. Using the self-intersection formulas, we obtain d=1 r 0 π d cr B d t r xd d! = exp d=1 r=0 S r d 1 d 1 κ r d t r xd d!. 27 To account for the alternating factor 1 d 1 and the κ subscript, we define the coefficients C d r by d=1 r 1 C r d t r xd d! = log 1 + d d=1 i=1 1 1 it 1 d t d x d d! The vanishing S d r<d 1 = 0 implies the vanishing C d r< 1 = 0. Again using Mumford s Grothendieck-Riemann-Roch calculation 14], c t E = i 1 B 2i 2i2i 1 κ 2i 1t 2i 1. Putting the above results together yields the following formula:. π d cr F d r d xd t d=1 r 0 exp d! = i 1 B 2i 2i2i 1 κ 2i 1t 2i 1 d=1 r 1 C r d κ r t r xd d!. The restrictions on g, d, and r in the statement of Proposition 4 are obtained from Extended relations The universal curve carries the basic divisor classes ɛ : U Q g P 1, d s = c 1 S U, ω = c 1 ω π 24

25 obtained from the universal subsheaf S U of the moduli of stable quotients and the ɛ-relative dualizing sheaf. Following 13, Proposition 5], we can obtain a much larger set of relations in the tautological ring of M g by including factors of ɛ s a i ω b i in the integrand: n ν ɛ s a i ω b i 0 c Q g P 1, d] vir = 0 in A M g, Q i=1 when c > 0. We will study the associated relations where the a i are always 1. The b i then form the parts of a partition σ. To state the relations we obtain, we start by extending the function γ of Section 3.1, γ SQ = i 1 + σ B 2i 2i2i 1 κ 2i 1t 2i 1 d=1 r= 1 Let γ SQ be defined by a similar formula, C d r κ r+ σ t r xd d! d lσ t σ p σ Autσ. γ SQ = i 1 + σ B 2i 2i2i 1 κ 2i 1 t 2i 1 d=1 r= 1 C d r κ r+ σ t r xd d! d lσ t σ p σ Autσ. The sign of t in t σ does not change in γ SQ. The κ 1 terms which appear will later be set to 0. The full system of relations are obtained from the coefficients of the functions exp γ SQ, exp κ r t r p r+1 exp γ SQ Theorem 5. In R r M g, the relation r=0 ]trxdpσ exp γ SQ = 1 g exp holds when g 2d 1 + σ < r. 25 r=0 ] κ r t r p r+1 exp γ SQ t r x d p σ

26 Again, we see the genus shifting mod 2 property. If the relation holds in genus g, then the same relation holds in genera h = g 2δ for all natural numbers δ N. In case σ =, Theorem 5 specializes to the relation ]trxd ] exp γt, x = 1 g exp γ t, x t r x ] d = 1 g+r exp γt, x t r x d, nontrivial only if g r + 1 mod 2. If the mod 2 condition holds, then we obtain the relations of Proposition 4. Consider the case σ = 1. The left side of the relation is then exp γt, x d=1 s= 1 C s d s+1 dxd κ s+1 t d! ] t r x d. The right side is 1 g exp γ t, x κ 0 t 0 + d=1 s= 1 C s d s+1 dxd ] κ s+1 t. d! t r x d If g r + 1 mod 2, then the large terms cancel and we obtain ] κ 0 exp γt, x = 0. t r x d Since κ 0 = 2g 2 and g 2d < r = g 2d 1 < r, we recover most but not all of the σ = equations. If g r mod 2, then the resulting equation is exp γt, x κ 0 2 C s d s+1 dxd ] κ s+1 t = 0 d! t r x d when g 2d < r. d=1 s= 1 26

27 3.5 Proof of Theorem Partitions, differential operators, and logs. We will write partitions σ as 1 n 1 2 n 2 3 n 3... with lσ = i n i and σ = i in i. The empty partition corresponding to is permitted. In all cases, we have Autσ = n 1!n 2!n 3!. In the infinite set of variables {p 1, p 2, p 3,...}, let Φ p t, x = σ d d=0 i=1 1 1 it 1 d d! x d t d d lσ t σ p σ Autσ, where the first sum is over all partitions σ. The summand corresponding to the empty partition equals Φt, x defined in 20. The function Φ p is easily obtained from Φ, Φ p t, x = exp p i t i x d Φt, x. dx i=1 Let D denote the differential operator D = i=1 p i t i x d dx. Expanding the exponential of D, we obtain Φ p = Φ + DΦ D2 Φ D3 Φ = Φ 1 + DΦ Φ + 1 D 2 Φ 2 Φ + 1 D 3 Φ 6 Φ Let γ = logφ be the logarithm, Dγ = DΦ Φ. 27

28 After applying the logarithm to 28, we see logφ p = γ + log 1 + Dγ D2 γ + Dγ = γ + Dγ D2 γ +... where the dots stand for a universal expression in the D k γ. remarkable simplification occurs, logφ p = exp p i t i x d γ. dx i=1 The result follows from a general identity. Proposition 7. If f is a function of x, then log exp λx d f = exp λx d logf. dx dx Proof. A simple computation for monomials in x shows exp λx d x k = e λ x k. dx Hence, since the differential operator is additive, exp λx d fx = fe λ x. dx The Proposition follows immediately. In fact, a We apply Proposition 7 to logφ p. The coefficients of the logarithm may be written as logφ p = C r d σ t r xd d! pσ σ d=1 r= 1 = C r d t r xd d! exp dp i t i = σ d=1 r= 1 d=1 r= 1 C d r t r xd d! i=1 d lσ t σ p σ Autσ We have expressed the coefficients C d r σ of logφ p solely in terms of the coefficients C d r of logφ. 28.

29 3.5.2 Cutting classes Let θ i A 1 U, Q be the class of the i th section of the universal curve ɛ : U C d g 29 The class s = c 1 S U on the universal curve over Q gp 1, d restricted to the C -fixed locus C d g/s d and pulled-back to 29 yields s = θ θ d A 1 U, Q. We calculate ɛ s ω b = ψ b ψ b d A b C d g, Q Wick form We repeat the Wick analysis of Section 3.3 for the vanishings ν l i=1 ɛ sω b i 0 c Q g P 1, d] vir = 0 in A M g, Q when c > 0. We start by writing a formula for exp p i t i ɛ sω i c r F d t r d=1 r 0 π d i=1 1 t d x d d!. Applying the Wick formula to equation 30 for the cutting classes, we see π d exp p i t i ɛ sω i c r F d t r 1 x d t d d! = exp γ SQ 31 d=1 r 0 i=1 where γ SQ is defined by B 2i γ SQ = 2i2i 1 κ 2i 1t 2i 1 + i 1 σ d=1 r= 1 C d r σκ r t r xd d! pσ. We follow here the notation of Section 3.5.1, Φ p t, x = σ d d=0 i=1 1 1 it 1 d d! x d t d d lσ t σ p σ Autσ, 29

30 logφ p = σ d=1 r= 1 C d r σ t r xd d! pσ. In the Wick analysis, the class ɛ sω b simply acts as dt b. Using the expression for the coefficents C r d σ in terms of C r d Section 3.5.1, we obtain the following result from 31. derived in Proposition 8. We have exp p i t i ɛ sω i c r F d t r d=1 r 0 π d i= Geometric construction 1 t d x d d! = exp γsq. We apply C -localization on Q g P 1, d to the geometric vanishing ν l i=1 ɛ sω b i 0 c Q g P 1, d] vir = 0 in A M g, Q 32 when c > 0. The result is the relation l π ɛ sω b i c g d 1+c F d + i=1 1 g d 1 l i=1 ɛ s 1ω b i ] g d 1+ i c F d b i+c = 0 33 in R M g. After applying the Wick formula of Proposition 8, we immediately obtain Theorem 5. The first summand in 33 yields the left side ] exp γ SQ t r x d p σ of the relation of Theorem 5. The second summand produces the right side 1 g exp r=0 ] κ r t r p r+1 exp γ SQ. 34 t r x d p σ 30

31 Recall the localization of the virtual class over P 1 is 1 g d 1 π d d! Of the sign prefactor 1 g d 1, c F d ] g d 1+c. 1 1 is used to move the term to the right side, 1 d is absorbed in the t of the definition of γ SQ, 1 g remains in 34. The 1 of s 1 produces the the factor exp r=0 κ rt r p r+1. Finally, a simple dimension calculation remembering c > 0 implies the validity of the relation when g 2d 1 + σ < r. 4 Analysis of the relations 4.1 Expanded form Let σ = 1 a 1 2 a 2 3 a 3... be a partition of length lσ and size σ. We can directly write the corresponding tautological relation in R r M g obtained from Theorem 5. A subpartition σ σ is obtained by selecting a nontrivial subset of the parts of σ. A division of σ is a disjoint union σ = σ 1 σ 2 σ of subpartitions which exhausts σ. The subpartitions in 35 are unordered. Let Sσ be the set of divisions of σ. For example, S = { , }, S1 3 = { 1 3, }. We will use the notation σ to denote a division of σ with subpartitions σ i. Let mσ 1 Autσ = Autσ lσ i=1 Autσi. 31

32 Here, Autσ is the group permuting equal subpartitions. The factor mσ may be interpreted as counting the number of different ways the disjoint union can be made. To write explicitly the p σ coefficient of expγ SQ, we introduce the functions F n,m t, x = C s d κ s+m t s+m dn x d d! for n, m 1. Then, Autσ ] exp γ SQ = t r x d p σ exp γt, x d=1 s= 1 σ Sσ lσ mσ Let σ, be a division of σ with a marked subpartition, i=1 F lσ i, σ i ] t r x d. σ = σ σ 1 σ 2 σ 3..., 36 labelled by the superscript. The marked subpartition is permitted to be empty. Let S σ denote the set of marked divisions of σ. Let mσ, = 1 Autσ Autσ Autσ lσ, i=1 Autσ i. The length lσ, is the number of unmarked subpartitions. Then, Autσ times the right side of Theorem 5 may be written as 1 g+ σ Autσ σ, S σ exp γ t, x lσ mσ, j=1 lσ, κ σ j 1 t σ j 1 i=1 F lσ i, σ t, x ] i t r x d To write Theorem 5 in the simplest form, the following definition using the Kronecker δ is useful, m ± σ, = 1 ± δ 0, σ mσ,. 32

33 There are two cases. If g r + σ mod 2, then Theorem 3 is equivalent to the vanishing of Autσ exp γ lσ lσ, m σ, κ σ j 1t σ j 1 ]. t r x d σ, S σ j=1 i=1 F lσ i, σ i If g r + σ + 1 mod 2, then Theorem 5 is equivalent to the vanishing of Autσ exp γ lσ lσ, m + σ, κ σ j 1t σ j 1 ]. t r x d σ, S σ j=1 i=1 F lσ i, σ i In either case, the relations are valid in the ring R M g only if the condition g 2d 1 + σ < r holds. We denote the above relation corresponding to g, r, d, and σ and depending upon the parity of g r σ by Rg, r, d, σ = 0 The Autσ prefactor is included in Rg, r, d, σ, but is only relevant when σ has repeated parts. In case of repeated parts, the automorphism scaled normalization is more convenient. 4.2 Further examples If σ = k has a single part, then the two cases of Theorem 5 are the following. If g r + k mod 2, we have exp γ κ k 1 t k 1 ] t r x d = 0 which is a consequence of the σ = case. If g r + k + 1 mod 2, we have exp γ κ k 1 t k 1 + 2F 1,k ] t r x d = 0 If σ = k 1 k 2 has two distinct parts, then the two cases of Theorem 5 are as follows. If g r + k 1 + k 2 mod 2, we have exp γ κ k1 1κ k2 1t k 1+k κ k1 1t k 1 1 F 1,k2 + κ k2 1t k 2 1 F 1,k1 ] t r x d = 0. 33

34 If g r + k 1 + k mod 2, we have exp γ κ k1 1κ k2 1t k 1+k κ k1 1t k 1 1 F 1,k2 + κ k2 1t k 2 1 F 1,k1 + 2F 2,k1 +k 2 + 2F 1,k1 F 1,k2 ] t r x d = 0. In fact, the g r + k 1 + k 2 mod 2 equation above is not new. The genus g and codimension r 1 = r k case of partition k 1 yields exp γ κ k1 1t k F 1,k1 ] t r 1 x d = 0. After multiplication with κ k2 1t k 2 1, we obtain exp γ κ k1 1κ k2 1t k 1+k κ k2 1t k 2 1 F 1,k1 ] t r x d = 0. Summed with the corresponding equation with k 1 and k 2 interchanged yields the above g r + k 1 + k 2 mod 2 case. 4.3 Expanded form revisited Consider the partition σ = k 1 k 2 k l with distinct parts. Relation Rg, r, d, σ, in the g r + σ mod 2 case, is the vanishing of exp γ σ, S σ since all the factors mσ, are 1. Rg, r, d, σ is the vanishing of exp γ σ, S σ lσ lσ, 1 δ 0, σ κ σ j 1t σ j 1 j=1 lσ 1 + δ 0, σ i=1 F lσ i, σ i ] t r x d In the g r + σ + 1 mod 2 case, j=1 lσ, κ σ j 1t σ j 1 i=1 F lσ i, σ i ] t r x d for the same reason. If σ has repeated parts, the relation Rg, r, d, σ is obtained by viewing the parts as distinct and specializing the indicies afterwards. For example, the two cases for σ = k 2 are as follows. If g r + 2k mod 2, we have exp γ κ k 1 κ k 1 t 2k 2 + 2κ k 1 t k 1 F 1,k ] t r x d = 0. 34

35 If g r + 2k + 1 mod 2, we have exp γ κ k 1 κ k 1 t 2k 2 + 2κ k 1 t k 1 F 1,k + 2F 2,2k + 2F 1,k F 1,k ] t r x d = 0. The factors occur via repetition of terms in the formulas for distinct parts. Proposition 9. The relation Rg, r, d, σ in the g r + σ mod 2 case is a consequence of the relations in Rg, r, d, σ where g r + σ + 1 mod 2 and σ σ is a strictly smaller partition. Proof. The strategy follows the example of the phenonenon already discussed in Section 4.2. If g r + σ mod 2, then for every subpartition τ σ of odd length, we have g r τ + lτ + σ/τ + 1 mod 2 where σ/τ is the complement of τ. The relation κ τi 1 Rg, r τ + lτ, d, σ/τ i is of codimension r. Let g r + σ mod 2, and let σ have distinct parts. The formula Rg, r, d, σ = 2 lτ+2 2 B lτ+1 κ τi 1 R g, r τ + lτ, d, σ/τ lτ + 1 τ σ i 37 follows easily by grouping like terms and the Bernoulli identity n 2 2k n+2 2 B 2k = B n k 1 2k n + 1 k 1 for n > 0. The sum in 37 is over all subpartitions τ σ of odd length. The proof of the Bernoulli identity 38 is straightforward. Let 2 i+2 2 a i = B i+1, Ax = i i=0 a i x i i!.

36 Using the definition of the Bernoulli numbers as we see Ax = 2 x 2 i x r 1B r i=0 x e x 1 = x i B i i!, r! = 2 x i=0 2x e 2x 1 x 2 = e x e x. The identity 38 follows from the series relation e x Ax = Ax 2. Formula 37 is valid for Rg, r, d, σ even when σ has repeated parts: the sum should be interpreted as running over all odd subsets τ σ viewing the parts of σ as distinct. 4.4 Recasting We will recast the relations Rg, r, d, σ in case g r + σ + 1 mod 2 in a more convenient form. The result will be crucial to the further analysis in Section 5. Let g r + σ + 1 mod 2, and let Sg, r, d, σ denote the κ polynomial Aut exp γt, x + F lσ, σ + δ lσ,1 p σ κ σ 1 ]. 2 Autσ t r x d p σ σ We can write Sg, r, d, σ in terms of our previous relations Rg, r, d, σ satisfying g r + σ + 1 mod 2 and σ σ: If g r + σ + 1 mod 2, then for every subpartition τ σ of even length including the case τ =, we have g r τ + lτ + σ/τ + 1 mod 2 where σ/τ is the complement of τ. The relation κ τi 1 Rg, r τ + lτ, d, σ/τ i 36

37 is of codimension r. In order to express S in terms of R, we define z i Q by 2 e x + e = x i z x i i!. i=0 Let g r + σ + 1 mod 2, and let σ have distinct parts. The formula Sg, r, d, σ = z lτ 2 κ lτ+1 τi 1 R g, r τ + lτ, d, σ/τ τ σ i 39 follows again grouping like terms and the combinatorial identity n zi i 2 i + 1 = z n n+1 2 n i 0 for n > 0. The sum in 39 is over all subpartitions τ σ of even length. As before, there the identity 40 is straightforward to prove. We see Zx = i=0 z i 2 i+1 x i i! = 1 e x/2 + e x/2. The identity 40 follows from the series relation e x Zx = e x/2 Zx. Formula 37 is valid for Sg, r, d, σ even when σ has repeated parts: the sum should be interpreted as running over all even subsets τ σ viewing the parts of σ as distinct. We have proved the following result. Proposition 10. In R r M g, the relation exp γt, x + F lσ, σ + δ lσ,1 p σ κ σ 1 ] = 0 2 Autσ t r x d p σ σ holds when g 2d 1 + σ < r and g r + σ + 1 mod 2. 37

38 5 Transformation 5.1 Differential equations The function Φ satisfies a basic differential equation obtained from the series definition: d d Φ tx dx dx Φ = 1 t Φ. After expanding and dividing by Φ, we find tx Φ xx Φ tφ x Φ + Φ x Φ = 1 t which can be written as t 2 xγ xx = t 2 xγ x 2 + t 2 γ x tγ x 1 41 where, as before, γ = logφ. Equation 41 has been studied by Ionel in Relations in the tautological ring 9]. We present here results of hers which will be useful for us. To kill the pole and match the required constant term, we will consider the function Γ = t i 1 The differential equation 41 becomes B 2i 2i2i 1 t2i 1 + γ txγ xx = xγ x tγ x The differential equation is easily seen to uniquely determine Γ once the initial conditions Γt, 0 = B 2i 2i2i 1 t2i i 1 are specified. By Ionel s first result, Γ x = x 2x + t 1 + 4x + k=1 j=0 k t k+1 q k,j x j 1 + 4x j k 2 1 where the postive integers q k,j defined to vanish unless k j 0 are defined via the recursion q k,j = 2k + 4j 2q k 1,j 1 + j + 1q k 1,j + 38 k 1 m=0 l=0 j 1 q m,l q k 1 m,j 1 l

39 from the initial value q 0,0 = 1. Ionel s second result is obtained by integrating Γ x with respect to x. She finds Γ = Γ0, x + t 4 log1 + 4x k t k+1 c k,j x j 1 + 4x j k 2 k=1 j=0 where the coefficients c k,j are determined by q k,j = 2k + 4jc k,j + j + 1c k,j+1 for k 1 and k j 0. While the derivation of the formula for Γ x is straightforward, the formula for Γ is quite subtle as the intial conditions given by the Bernoulli numbers are used to show the vanishing of constants of integration. Said differently, the recursions for q k,j and c k,j must be shown to imply the formula c k,0 = B k+1 kk + 1. A third result of Ionel s is the determination of the extremal c k,k, c k,k z k 6k! z k = log. 2k!3k! 72 k=1 k=1 The formula for Γ becomes simpler after the following very natural change of variables, t u = and y = x 1 + 4x 1 + 4x. 43 The change of variables defines a new function The formula for Γ implies Γu, y = Γt, x. 1 t Γu, y = 1 t Γ0, y 1 4 log1 + 4y k=1 j=0 k c k,j u k y j. 44 Ionel s fourth result relates coefficients of series after the change of variables 43. Given any series P t, x Qt, x]], 39

40 let P u, y be the series obtained from the change of variables 43. Ionel proves the coefficient relation P t, x ]t r x d = 1 d 1 + 4y r+2d 2 2 P u, y ] u r y d. 5.2 Analysis of the relations of Proposition 4 We now study in detail the simple relations of Proposition 4, exp γ ] t r x d = 0 R r M g when g 2d 1 < r and g r + 1 mod 2. Let γu, y = γt, x be obtained from the variable change 43. Equations 21, 42, and 44 together imply γu, y = κ 0 4 log1 + 4y + k=1 j=0 k κ k c k,j u k y j modulo κ 1 terms which we set to 0. Applying Ionel s coefficient result, exp γ = ]t 1 + 4y r+2d 2 r x d 2 exp γ ] u r y d = 1 + 4y r+2d 2 κ exp = 1 + 4y r g+2d 1 2 exp k=1 j=0 k=1 j=0 ] k κ k c k,j u k y j ] k κ k c k,j u k y j u r y d u r y d. In the last line, the substitution κ 0 = 2g 2 has been made. Consider first the exponent of 1 + 4y. By the assumptions on g and r in Proposition 4, r g + 2d and the fraction is integral. Hence, the y degree of the prefactor 1 + 4y r g+2d

41 is exactly r g+2d 1. The y degree of the exponential factor is bounded from 2 above by the u degree. We conclude ] 1 + 4y r g+2d 1 2 exp is the trivial relation unless k=1 j=0 r d r g + 2d 1 2 k κ k c k,j u k y j = r 2 + g u r y d = 0 Rewriting the inequality, we obtain 3r g +1 which is equivalent to r > g 3. The conclusion is in agreement with the proven freeness of R M g up to and including degree g 3. A similar connection between Proposition 4 and Ionel s relations in 9] has also been found by Shengmao Zhu 21]. 5.3 Analysis of the relations of Theorem 5 For the relations of Theorem 5, we will require additional notation. To start, let k γ c u, y = κ k c k,j u k y j. By Ionel s second result, k=1 j=0 1 t Γ = 1 t Γ0, x log1 + 4x k=1 j=0. k t k c k,j x j 1 + 4x j k Let c 0 k,j = c k,j. We define the constants c n k,j for n 1 by x d n 1 dx t Γ = x d n 1 1 dx 2t x 2t k+n t k c n k,j x j 1 + 4x j k 2. k=0 j=0 41

42 Lemma 11. For n > 0, there are constants b n j satisfying x d n 1 1 n x = b n j u 1 y j. dx 2t Moreover, b n n 1 = 2 n 2 2n 5!! where 1!! = 1 and 3!! = 1. Proof. The result is obtained by simple induction. The negative evaluations 1!! = 1 and 3!! = 1 arise from the Γ-regularization. j=0 Lemma 12. For n > 0, we have c n 0,n = 4 n 1 n 1!. Proof. The coefficients c n 0,n are obtained directly from the t 0 summand 1 4 log1+ 4x of 45. Lemma 13. For n > 0 and k > 0, we have c n k,k+n = 6k6k + 4 6k + 4n 1 c k,k. Proof. The coefficients c n d k,k+n are extremal. The differential operators x dx must always attack the 1 + 4x j k 2 in order to contribute c n k,k+n. The formula follows by inspection. Consider next the full set of equations given by Theorem 5 in the expanded form of Section 4. The function F n,m may be rewritten as F n,m t, x = d=1 s= 1 = t m x d dx C d s κ s+m t s+m dn x d n d=1 s= 1 d! C d s κ s+m t s xd d!. We may write the result in terms of the constants b n j and c n k,j, t m n F n,m = δ n,1 κ m y n 2 n 1 κ m 1 b n j u n 1 y j j=0 k+n κ k+m c n k,ju k+n y j k=0 j=0 42

43 Define the functions G n,m u, y by n 1 k+n G n,m u, y = κ m 1 b n j u n 1 y j κ k+m c n k,ju k+n y j. j=0 k=0 j=0 Let σ = 1 a 1 2 a 2 3 a 3... be a partition of length lσ and size σ. assume the parity condition We g r + σ Let G ± σ u, y be the following function associated to σ, G ± σ u, y = σ Sσ lσ i=1 G lσ i, σ i ± δ lσ i, y κ σ i 1. The relations of Theorem 5 in the the expanded form of Section 4.1 written in the variables u and y are 1 + 4y r σ g+2d 1 2 exp γ c G + σ + G σ ] u r σ +lσ y d = 0 In fact, the relations of Proposition 10 here take a much more efficient form. We obtain the following result. Proposition 14. In R r M g, the relation 1 + 4y r σ g+2d 1 2 exp γ c σ G lσ, σ holds when g 2d 1 + σ < r and g r + σ + 1 mod 2. p σ ] = 0 Autσ u r σ +lσ y d p σ Consider the exponent of 1 + 4y. By the inequality and the parity condition 46, r σ g + 2d and the fraction is integral. Hence, the y degree of the prefactor 1 + 4y r σ g+2d

44 is exactly r σ g+2d 1 2. The y degree of the exponential factor is bounded from above by the u degree. We conclude the relation of Theorem 4 is trivial unless r σ + lσ d Rewriting the inequality, we obtain r σ g + 2d 1 2 3r g σ 2lσ = r σ 2 + g which is consistent with the proven freeness of R M g up to and including degree g Another form A subset of the equations of Proposition 14 admits an especially simple description. Consider the function H n,m u = 2 n 2 2n 5!! κ m 1 u n n 1 n 1! κ m u n + 6k6k + 4 6k + 4n 1c k,k κ k+m u k+n. k=1 Proposition 15. In R r M g, the relation exp c k,k κ k u k p σ H lσ, σ ] = 0 Autσ u r σ +lσ p σ k=1 σ holds when 3r g σ 2lσ and g r + σ + 1 mod 2. Proof. Let g r + σ + 1, and let 3 2 r 1 2 g σ + lσ = > 0. 2 By the parity condition, δ is an integer. For 0 δ, let E δ g, r, σ = exp γ c + p σ G lσ, σ ]. Autσ u r σ +lσ y r σ +lσ δ p σ σ. 44