arXiv:1912.07720v1 [math.AG] 16 Dec 2019 Boundary Expression for Chern Classes of the Hodge Bundle on Spaces of Cyclic Covers Bryson Owens and Seamus Somerstep Abstract We compute an explicit formula for the first Chern class of the Hodge Bundle over the space of admissible cyclic Z3Z covers of n-pointed rational stable curves as a linear combination of boundary strata. We then apply this formula to give a recursive formula for calculating certain Hodge integrals containing λ1 . We also consider covers with a Z2Z action for which we compute λ2 as a linear combination of codimension two boundary strata. 1 Introduction This paper studies the intersection theory of moduli spaces of cyclic admissible covers. We extend a result about the first Chern class of the Hodge bundle over spaces of degree two admissible covers of nodal stable curves, which first appeared in an unpublished undergraduate honors thesis [6] and two years later in [1], to the case of cyclic degree three covers. We begin with the Deligne-Mumford compactification of the moduli space of T -pointed curves of genus 0, M0,T . Let ω = (123), ω = (132) ∈ Z3Z. We then consider the moduli space Adm 3 of admissible genus g covers of g→ − 0(n|m) (n + m)-pointed genus 0 curves which ramify over the marked points of the covered curve, such that n branch points have monodromy ω and m branch points have monodromy ω. For notational simplicity, we denote this space as Admg(n|m) , where the genus of the curve being covered is always 0 and the degree of the cover is understood in context (the degree shall be 3 throughout this paper except in § 4, where we consider covers of degree 2). We denote by λi the i-th Chern class of the Hodge Bundle over Admg(n|m) and by Dij the sum of all irreducible codimension one boundary strata parameterizing nodal covers with i branch points of monodromy ω and j branch points with monodromy ω on one component. The branch morphism br : Admg(n|m) → M0,T is a bijection on the points of the moduli spaces, but has degree due to the fact that every admissible cover has a cyclic automorphism group of order three. From [3], we know that the Chow ring A∗ (M0,T ) is generated by its boundary strata. From this it follows that λ1 , a codimension one tautological class of Admg(n|m) , can be expressed as the pullback via the branch morphism of a linear combination of boundary divisors of M0,T . Our main result in this paper gives an explicit formula for computing the coefficients of this linear combination. P Theorem 1.1. λ1 can be expressed as 3π ∗ ( αji Dij ) over all symmetrized boundary divisors Dij of M0,T , where αji =     −i−j)   2(i+j)(T 27(T −1)    2(i+j−1)(T −i−j−1)   27(T −1) i − j ≡ 0 mod 3 i − j ≡ ±1 mod 3 We prove this theorem by computing for any curve γ in M0,T π∗ (λ1 ) · γ and P αji Dij · γ using the formula described in Theorem 1.1 and verify these two expressions are always equal. As an application of this formula we give the following recursive method for calculating certain Hodge integrals containing λ1 . Theorem 1.2. The family of Hodge integrals Z λn+m−3 1 Admg(n|m) n+m−3 Admg(n|m) λ1 R is given by the recursive formula:     Z Z m n X X 2(i + j − 1)(T − i − j − 1) n + m − 3 n m i+j−2 =3 λ1 λn+m−i−j−2 1 9(T − 1) i+j−2 i j Adm(i+1|j) Adm(n−i|m−j+1) i=0 j=0 This theorem follows naturally from Theorem 1.1 as well as how the Hodge bundle, and consequently its Chern classes, split. 2 We also study the second Chern class, λ2 , in the case of degree two admissible cover. Using Mumford’s relations and the expression for λ1 in [1] we prove the following theorem. Theorem 1.3. Let ∆i1 ,i2 ,i3 denote the codimension 2 symmeterized stratum in Admg with i1 branch points on the left component, i2 branch points on the middle component, and i3 branch points on P the right component. Then λ2 = αi1 ,i2 ,i3 ∆i1 ,i2 ,i3 , where     i1 i2 i3 (2i1 i2 +2i1 i3 +2i2 i3 −i1 −2i2 −i3 )    32(i1 +i2 +i3 −1)(i1 +i2 −1)(i2 +i3 −1)     αi1 ,i2 ,i3 = (i1 −1)(i2 )(i3 −1)((i2 +i3 −1)(i1 +i2 )+(i1 +i2 −1)(i2 +i3 ))  32(i1 +i2 +i3 −1)(i1 +i2 )(i2 +i3 )        (i −1)(i2 +i3 −1)(i2 +1)(i3 )(i1 +i2 −1)+(i3 )(i1 +i2 )(i2 −1)(i1 −1)(i2 +i3 )   1 32(i1 +i2 +i3 −1)(i2 +i3 )(i1 +i2 −1) i1 , i2 , i3 ≡ 0 mod 2 i1 , i3 ≡ 1, i2 ≡ 0 mod 2 i1 , i2 ≡ 1, i3 ≡ 0 mod 2 Acknowledgements The authors thank Dr. Renzo Cavalieri for suggesting the problem and for his helpful guidance. They also thank Adam Afandi for his help in teaching us the intersection theory of M0,T 2 2.1 Preliminaries The Moduli Space of Pointed Stable Rational Curves M0,n We denote by M0,n the moduli space of n marked points on P1 (C), up to the action of PGL(2,C). That is, each point on M0,n corresponds to an isomorphism class of n distinct marked points on P1 (C). The theory of Möbius transformations tells us that there exists a unique automorphism on P1 (C) that sends any 3-tuple of points to any other 3-tuple of points. That is, given an ntuple p1 , ... pn , there exists a unique Möbius transformation Φ : P1 (C) → P1 (C) such that Φ(p1 ) = 0, Φ(p2 ) = 1, Φ(p3 ) = ∞ (these numbers are chosen simply by convention and could be chosen to be any other three points in P1 (C)) and the other marked points are determined uniquely as the images by Φ of p4 , ..., pn . Since any P1 (C) marked by some n-tuple is isomorphic to a P1 (C) 3 marked with an n-tuple whose first three coordinates are 0, 1, ∞ by some Möbius Transformation, we may pick the latter copy of P1 (C) to be the representative for the isomorphism class. Therefore, since each isomorphism class has equivalent first three points, we may parametrize them by their remaining n − 3 points. Therefore, M0,n is n − 3 complex dimensional. For example, M0,3 is a single point since all copies of P1 (C) marked with three points are isomorphic to P1 (C) marked with 0, 1, ∞. While M0,n is not a compact topological space, it admits a compactification known as the Deligne-Mumford compactification, denoted M0,T , which parameterizes nodal stable curves. Definition 2.1. A nodal stable curve is a tree of projective lines, which have the following properties: • Components of the tree are copies of P1 (C) connected at nodes. • There are no closed circuits. • Each component has at least three special points (marked points or nodes). The complement of M0,n in M0,T , called the boundary of M0,T , is the set of points parameterizing marked stable nodal curves. This boundary is stratified by topological type, with each stratum being indexed by a dual graph, which we now define. Definition 2.2. Given a nodal stable curve C dual graph is a connected graph with the following properties: • Each vertex corresponds to a copy of P1 (C). • Half-edges connected to a vertex correspond to marked points on the component corresponding to that vertex. • Edges between vertices correspond to nodes connecting the components corresponding to the vertices. 4 A given dual graph only represents how each component is connected to the others and which marked points are on which components. Therefore, dual graphs represent strata (loci) of points rather than a single boundary point. We use this combinatorial representation for calculations in this paper. As an example, consider the nodal curve consisting of two copies of P1 (C) each containing two marked points. This curve is represented by the dual graph in Figure 1 and parametrizes a point in the boundary of M0,4 (the other two points in the boundary of M0,T are parametrized by dual graphs with the same structure, with the marked points relabeled): p1 p3 p2 p4 Figure 1: Boundary Point in M0,4 Definition 2.3. Consider M0,n+m and the action of Sn ×Sm where the element in Sn permutes the first n points and the element in Sm permutes the last m points. Then, a symmetrized stratum is the orbit of a stratum via this action. For example, consider Figure 1. The other two boundary points in M0,4 have dual graphs with the same structure, but with the marked points relabeled as follows: p1 p2 p3 p4 or Then, there is a single symmetrized strata which is the union of the three boundary points above. 5 p1 p3 p4 p2 We denote this symmetrized stratum as an unlabeled copy of the dual graphs of the boundary points in the symmetrized stratum. In the example of boundary points of M0,4 this is shown in Figure 2 Figure 2: Symmetrized Stratum in M0,4 The codimension 1 boundary strata of M0,T are called boundary divisors. These are especially important for us as π∗ (λ1 ) is expressible as a linear combination of boundary divisors of M0,T . A nice combinatorial representation of the codimension of boundary strata is the number of edges in the dual graph of the stratum. For example, a boundary divisor has one edge connecting two vertices, each with a certain number of half-edges connected to it. 2.2 The Chow Ring A∗ (M0,T ) The Chow Ring of M0,T is a ring whose elements are equivalence classes of subvarieties of M0,T , graded by codimension. Ai (M0,T ) is then the section of the ring corresponding to codimension i subvarieties of M0,T . From Keel [3], we know that A∗ (M0,T ) is generated by the boundary strata of M0,T . The binary operations on A∗ (M0,T ) are formal addition and a multiplication, which corresponds to intersection of strata. Naively, we may expect this multiplication to simply be set theoretic intersection, however, as we will see, this is not always the case. To see why, note that 6 since A∗ (M0,T ) is a graded ring, the multiplication must be a map · : Ai (M0,T ) × Aj (M0,T ) → Ai+j (M0,T ) That is, codim(X · Y ) = codim(X) + codim(Y ). For an easy example of where set theoretic intersections fail to satisfy this codimension requirement, consider a self intersection. Obviously, X ∩ X = X, so the codimensions can only follow the above condition if X = M0,T . Therefore, we must consider two types of intersections: transverse and non-transverse, where multiplication on the Chow Ring is set theoretic intersection if and only if the intersection is transverse. If the intersection is non-transverse, which we define in sec 2.4.1, then we must algebraically deform one of the subvarieties so that the intersection respects codimension. In order to actually compute these non-transverse intersections, we require the use of ψ-classes. 2.3 Chern Classes Let E be a complex rank r vector bundle. The i-th Chern class, ci (E) is a codimension i characteristic class of E, which has the following properties [4] which we use throughout this paper: • c0 (E) = 1 • ci (E) = 0 for i > r • c1 (E ⊕ F ) = c1 (E) + c1 (F ) 2.4 ψ-Classes In order to fully understand the intersection theory of M0,T we must first introudce the ideas of ψ-classes. First, note that M0,T admits sections in its universal family. We have a sheaf on the universal family whose stalks are differential 1-forms on the curve parametrized. This is called the relative dualizing sheaf, and is denoted ωπ . This gives a means to define the i-th ψ-class 7 Definition 2.4. The i-th ψ-class, denoted ψi is the first Chern class of the restriction of ωπ to the i-th section of M0,T ψi = c1 (s∗i (ωπ )) There is an important lemma that is helpful in computing ψ classes on M0,T . It states that ψi on M0,T is equal to the sum of all boundary divisors where the i-th point is fixed on the left twig and two other marked points are fixed on the right twig. For our work the two important computations are that ψi on M0,3 is zero (there is no way to distribute three points to obtain a stable boundary divisor) and ψi on M0,4 is the class of a point (there is only one way to distribute the last point so the dual graph is stable). 2.5 Intersections of Strata Take two strata in M0,T with dual graphs Γ1 , Γ2 . In order to calculate the intersection of the two strata, we must find the minimal refinement of the graphs Γ1 , Γ2 . A refinement is a graph Γ such that both Γ1 , Γ2 can be obtained by contracting edges. Color the edges which are not contracted to obtain Γ1 red and the edges which are not contracted to obtain Γ2 blue. The refinement is said to be minimal if all edges are colored red, blue, or both. This method of coloring edges which are not contracted to obtain the original dual graphs from the refinement give a simple test for whether an intersection is transverse or non-transverse. The intersection is transverse if no edge in the minimal refinement is colored both red and blue, and the minimal refinement Γ is dual to the product of the strata to which Γ1 , Γ2 are dual. Furthermore, an intersection is non-transverse if there exists an edge in Γ which is colored both red and blue. This bicolored edge is called a common edge and corresponds to curves parametrized by the two strata having a common node. We can now use ψ-classes to compute non-transverse intersections. 8 2.5.1 Non-Transverse Intersections Let S be a boundary divisor of M0,n , T1 be the set of marked points on one component, and T2 the set of marked points on the other component. Then, S ∼ = M0,T1 ∪{·} × M0,T2 ∪{⋆} where ·, ⋆ are the two marked points which, when glued together, form the node of the curve parametrized by S. Then, there exist projections as shown in Figure 3. S∼ = M0,T1 ∪{·} × M0,T2 ∪{⋆} i M0,T ρ2 ρ1 M0,T1 ∪{·} M0,T2 ∪{⋆} Figure 3: Splitting a boundary divisor into irreducible components We can then define the notion of adding −ψ-classes at half-edges · and ⋆ as follows: −ψ· − ψ⋆ := i∗ (ρ∗1 (−ψ· ) + ρ∗2 (−ψ⋆ )) In order to compute a non-transverse intersection, we use this idea of adding −ψ-classes at the common edge. That is, say two strata S1 , S2 intersect non-transversally and let the refinement of the two graphs dual to S1 , S2 be Γ. Let the bicolored edges of Γ break into half-edges ·i , ⋆i by projections ρ1 , ρ2 . Then, S1 · S2 = Y −ψ·i − ψ⋆i over each common edge, supported on the stratum to which Γ is dual. 2.6 The Space of Cyclic Admissible Covers Definition 2.5. A degree d cyclic admissible cover is a curve C along with a map π : C → X of degree d which satisfies the following: • C has an action by the cyclic group of order d, and π is the quotient map. 9 • π is ètale everywhere except at a finite set of points, called the branch locus. That is, except at branch points (points in the branch locus), the fiber consists of d points. The fiber of a branch point will always consist of fewer than d points. • Each branch point has a mondromy representation in the cyclic group of degree d . • C and X are nodal curves, and the image of a node in C by π is a node in X. • Over a node, locally in analytic coordinates, X, C, π are described as follows, for some positive integer r not larger than d: – C : c1 c2 = a – X : x1 x2 = ar – x1 = cr1 , x2 = cr2 For our main results we wish to look at the moduli space of admissible covers of a marked rational curve, which ramify over the marked points. For this paper, we are focusing on the case of degree two and degree three covers. In the degree two case, the group Z2Z acts on C, meaning each branch point is fully ramified. In the degree three case we have a Z3Z action on the admissible covers. Note that Z3Z ∼ = h(123)i ≤ S3 . This is why each branch point has monodromy (123), or (132), which we denote as ω and ω, respectively. We denote the moduli space parameterizing admissible covers with n ω points and m ω points as Admg(n|m) , which has dimension n + m − 3 and parametrizes curves of genus n + m − 2. Finally, note that there exists a bijection from Admg(n|m) → M0,n+m called the branch morphism. The branch morphism has degree 1 d since there are d automorphisms of any admissible cover. Remark 2.1. Let ∆ be a boundary divisor in Admn|m . Then, ∆ ∼ = d(Admn1 |m1 × Admn2 |m2 ) for 10 some n1 , n2 , m1 , m2 [2]. We call this factor of d the gluing factor. Remark 2.2. We denote the space of genus g admissible cyclic degree three covers with n points with monodromy ω, m points with monodromy ω, and l unramified marked points as Admg(n|m|l) . We denote by Dij the symmetrized boundary divisor in M0,T whose pullback via π has i ω points and j ω points on one component. 2.7 The Hodge Bundle The Hodge bundle Eg is a complex rank g vector bundle over Admg , where the fiber of a curve is the vector space of one-forms. We denote λi := ci (Eg ). Along with the properties of Chern classes given earlier, the Chern classes of the Hodge bundle also satisfy Mumford’s relation, which states that (1 + λ1 + λ2 + λg )(1 − λ1 + λ2 − . . . ± λg ) = 1 [5]. Another important concept which is important for our result is the projection formula. Given the branch morphism π : Admg → M0,T , the projection formula states that for any curve C in M0,T , Z π∗ (λ1 ) = C Z λ1 . π∗ (C) Let Dij , be a boundary divisor in Admg(n|m|l) . Then, Dij ∼ = Admg1 (n1 |m1 |l1 ) × Admg2 (n2 |m2 |l2 ) . Then, we consider the Hodge bundle restricted to Dij . Case 1, the monodromies at the the marked points connected at the node are unramified: Eg |Dj ∼ = Eg1 ⊕ Eg2 i Case 1, the monodromies at the the marked points connected at the node are ramified: Eg |Dj ∼ = Eg1 ⊕ Eg2 ⊕ O 2 i 11 where O is the trivial line bundle of C. In either case, the following relation between Chern classes of the Hodge bundle and Chern R classes of the Hodge bundle restricted to a divisor can be derived. Let λL 1 , λ1 denote λ1 restricted to the left and right component of the divisor D respectively. Let the notation for the fundamental class be identical. Then, R R R L L 1 + λ1 + λ2 + · · · + λg = (1 + λL 1 + λ2 + · · · + λg )(1 + λ1 + λ2 + · · · λg ) This is used to compute λ1 of the Hodge bundle restricted to a boundary divisor. Namely, R L λ1 = [1]L λR 1 + [1] λ1 . This result generalizes to the Hodge Bundle restricted to boundary strata of higher codimension. In this case, λ1 be equal to the sum of λ1 restricted to each component of the boundary stratum (multiplied by a fundamental class on every other component). 3 The First Chern Class of the Hodge Bundle over Spaces of Z3Z Covers Remark 3.1. Let C be a boundary curve in Admn|m|l . Then, C ∼ = Admn1 |m1 |l1 × ... × Admnk |mk |lk where k = dim(Admn|m|l ) − 1. Since C is a curve, there is a single component, C4 for which ni + mi + li = 4 (for all other components, this sum equals three). This component is represented by a four-valent vertex in the graph dual to C. The graph dual to C is shown in Figure 4. Let ρ : C → C4 be the projection from C onto this four-valent component. Denote by Mi the monodromies of the marked points of C4 . For simplicity in the proof of the main theorem, we will refer to these monodromies as being at the edge on the dual graph 12 A D M2 M1 M4 M3 B C Figure 4: Graph Dual to a Stratum in Admn|m|l , A, B, C, D represent trivalent trees or single legs D I II III IV A A A A ω ω ω C ω B D ω ω ω e B D ω e ω e C B D e e C e e B C Figure 5: Pullbacks of boundary strata in M0,T via the branch morphism corresponding to the node connecting the marked point on the four-valent component to either A, B, C, or D; for example, in Figure 4 we would say the monodromy M2 is the monodromy at the edge connecting A to the four-valent component. Lemma 3.1. Every 1 dimensional boundary stratum in M0,T pulls back via the branch morphism to one of the four families in Figure 5. Proof. The stratum in M0,T corresponding to a trivalent tree has the highest possible codimension and so is of dimension zero. Making a single node 4-valent decreases the codimension by one and thus creates a curve. Pulling this curve back via the forgetful morphism introduces the monodromy of each marked point and thus gives us different cases, depending on the monodromy of each marked point. The monodromy of each branch point on the four-valent component is determined by the congruence of the number of ω points minus the number of ω points on the trivalent tree or single leg attached to the four-valent component at that node. That is, let iA , iB , iC , iD denote the number 13 of ω points on A, B, C, D, respectively and jA , jB , jC , jD the number of ω points on A, B, C, D, respectively. Then, the edge on the four-valent vertex connected to A will have monodromy e if iA − jB ≡ 0 mod 3, ω if iA − jA ≡ 1 mod 3 and ω if iA − jA ≡ −1 mod 3, and likewise for B, C, D. Since the product of the monodromies at all branch points must equal e, the sum of all the congruences iA − jA + iB − jB + iC − jC + iD − jD must be congruent to 0 mod 3. There are only four possible ways to add four terms modulo 3 to be congruent to 0 modulo 3, which correspond to the four families above. Here we consider the case where there are three ω points and and identity at the nodes and the cases with three ω points and an identity as the same family (Family II). Lemma 3.2. λ1n+m−2+k = 0 over Admn|m|l for any k ∈ Z≥0 Proof. Consider the forgetful morphism f : Admn|m|l → Admn|m , which forgets each unramified marked point. Then, [λ1 ]Admn|m|l = f ∗ ([λ1 ]Admn|m ) ⇒ ([λ1 ]Admn|m|l )n+m−2 = f ∗ ([λ1 ]Admn|m )n+m−2 ) Since the dimension of Admn|m = n + m − 3 and the codimension of ([λ1 ]Admn|m )n+m−2+k = n + m − 2 + k > n + m − 2 for any k ∈ Z≥0 , ([λ1 ]Admn|m )n+m−2+k = 0 ⇒ ([λ1 ]Admn|m|l )n+m−2+k = f ∗ (0) = 0 Lemma 3.3. Let ∆ be a boundary stratum in Admn|m . Then, there exist a series of projections as shown in Figure 6 Then, [λ1 ]Admn|m = P ρ∗i ([λ1 ]Admni |mi |li ). We say λ1 splits over the components of ∆. Proof. This comes by taking an iterated application of the process described in Figure 3. 14 ∆∼ = Adm(n1 |m1 |l1 ) × . . . × Adm(nk |mk |lk ) ρk ρ1 Adm(n1 |m1 |l1 ) ... Adm(nk |mk |lk ) Figure 6: λ1 splitting over components of a boundary stratum Lemma 3.4. Let γ be a boundary curve in M0,T . Then, π∗ (λ1 ) · γ = 0 for curves which pull back to Families II, III and IV and π∗ (λ1 ) · γ = 2 9 for curves which pull back to Family I from Lemma 3.1. Proof. By Lemma 3.3, π ∗ (γ) is isomorphic to a product of admissible cover spaces. Since all but one of these has exactly three marked points, they are each 1 3 the class of a point, since each space has three non-trivial automorphisms. However, for each edge, we must also multiply by the gluing factor of 3. Denote the four-valent component as C4 and let V be the set of tri-valent vertices in the graph dual to γ and E be the set of edges in the dual graph. Therefore, Y 1 3 λ1 · π ∗ (γ) = Z λ1 · = Z λ1 since |E| = |V | for a one-dimensional stratum C4 Y E 3· V 6=C4 C4 This integral is 0 if γ pulls back to Families II, III, or IV by Lemma 3.2 and 2 9 if γ pulls back to Family I, as a result of a computation in [2]. Remark 3.2. If π is a degree d map, π ∗ π∗ (λ1 ) = dλ1 . P Theorem 3.1. λ1 over Adm(n|m) can be expressed as 3π ∗ ( αji Dij ) over all symmetrized boundary divisors Dij of M0,T where αji =     −i−j)   2(i+j)(T 27(T −1)    2(i+j−1)(T −i−j−1)   27(T −1) 15 i − j ≡ 0 mod 3 i − j ≡ ±1 mod 3 A D B C Figure 7: Arbitrary Curve in M0,T Proof. Note: the coefficient in front of each symmetrized boundary divisor depends only on the sum of the ω and ω points on one component. Therefore, we can denote αji as αi+j , and if we let t1 = i + j denote the number of marked points on one component and t2 = T − i − j the number of marked points on the other component, we can express     2t1 t2   27(T monodromy at node: e −1) αt1 =    2(t −1)(t2 −1)  monodromy at node: ω, ω  127(T −1) We use this notation in the proof for simplicity. By Remark 3.3, the statement of this theorem is equivalent to saying that π∗ λ1 = P αji Dij . Therefore, to prove this theorem, we show that for every one-dimensional boundary stratum γ ∈ M0,T , γ · π ∗ λ1 = γ · X αji Dij (1) The dual graph of a one-dimensional boundary stratum contains a single four-valent vertex with each edge connected to either a trivalent tree or to a single leg (that is, it is a half-edge representing a marked point). We denote this as in Figure 7. Let γ be a curve in M0,T and let ta , tb , tc , td be the total number of marked points in A, B, C, D, respectively, where A, B, C, D are all either trivalent trees or single half-edges (if tA , tB , tC , tD = 1, respectively). For this proof, we calculate X αi+j γ · Dij 16 (2) and show that γ · π∗ (λ1 ) is equal to (2). By Lemma 3.3, λ1 over π ∗ (γ) is determined uniquely by λ1 on the four-valent component, so we only need to prove the four cases where γ pulls back to each of the four Families described in Lemma 3.1. It is easy to see that the only non-zero intersections between γ and the sum of symmetrized boundary divisors are the non-transverse intersections with DA , DB , DC , DD and the transverse intersections with DA∪B , DA∪C , DA∪D , where DX represents the boundary divisor with all the points in X on one component and all other points on the other component. For example, DA∪B would be represented by the dual graph in Figure 8. The y1 y2 .. . yt1 −1 yt1 x1 x2 .. . xt2 −1 xt2 where xi are the marked points in A ∪ B and yi are the marked points in C ∪ D Figure 8: The graph dual to the divisor DA∪B intersections with the divisors DA∪B , DA∪C , DA∪C are supported on the zero-dimensional boundary divisor which adds a single edge separating the points on A ∪ B, A ∪ C, or A ∪ D from all other points, respectively. For example, the intersection between γ and the divisor DA∪B is supported by the dual graph in Figure 9. Since this stratum is the product of a stratum of codimension one and a stratum of codimension equal to the dimension of M0,T −1, it is a stratum of dimension 0. Therefore, γ · DA∪B is the class of a point and thus contributes 1 to the overall sum in (2). The intersections A C B D Figure 9: γ · DA∪B 17 with DA , DB , DC , DD are non-transverse intersections. These each have a common node with γ, connecting the four-valent component to a trivalent component. Therefore, intersecting these two strata reduces to computing −ψn1 − ψn2 where n1 , n2 once glued together by pulling back via the respective projections form the node connecting the two components. Since the common node is between an M0,3 and an M0,4 , the intersection is computed by −[ψn1 ]M0,3 − [ψn1 ]M0,4 = 0 − 1 = −1. This means that each intersection of the curve γ with a divisor DA , DB , DC , DD contributes −1 to the overall sum in (2). Therefore, verifying (1) reduces to showing that γ · π∗ (λ1 ) = γ · (αA DA + αB DB + αC DC + αD DD + αA∪B DA∪B + αA∪C DA∪C + αA∪D DA∪D ) = −αA − αB − αC − αD + αA∪B + αA∪C + αA∪D (3) Note: if any of A, B, C, D are single half-edges rather than trivalent trees, the divisor DA , DB , DC , or DD is not a valid boundary divisor given our compactification of M0,n as its dual graph would not be stable. However, this problem resolves itself combinatorially as the coefficient in front of these divisors becomes zero since 1 − 0 ≡ 1 mod 3 so αX = (tX − 1)(. . .) = (1 − 1)(. . .) = 0 for whichever set X is a single half-edge. We now verify (3) for each of the four cases described in Figure 4. A Case I D ω ω ω C 18 ω B First, we determine which of the two cases each coefficients αA , αB , αC , αD , αA∪B , αA∪C , αA∪D fall into. We do this by using the monodromy at each node to determine iX − jX mod 3 for X any of the relevant sets. For example, the monodromy at the node of the four valent vertex connected to A is of type ω. Therefore, since the difference in ω and ω points must be congruent to 0 mod 3, iA + 1 − jA ≡ 0 mod 3. Therefore, iA − jA ≡ −1 mod 3. Doing this for each set gives us the following table: Coefficient αA αB αC αD αA∪B αA∪C αA∪D Equivalence mod 3 -1 -1 1 1 1 0 0 π ∗ λ1 · C = 2 9 by Lemma 3.4. ⇒ 2 2 = (−(tA − 1)(tB + tC + tD − 1) − (tB − 1)(tA + tC + tD − 1) 9 27(T − 1) − (tC − 1)(tA + tB + tD − 1) − (tD − 1)(tA + tB + tC − 1) + (tA + tB − 1)(tC + tD − 1) + (tA + tC )(tB + tD ) + (tA + tD )(tB + tC )) This equality is easily verifiable. Therefore, all curves of the form described above intersect correctly with π∗ λ1 . A Case II D A ω ω ω e B or D ω ω ω e C B C We prove the case that each monodromy is an ω point for this proof. The proof for 3 ω points is the same, except the equivalencies mod 3 change between 1 and -1. This does not change the proof since being congruent to ±1 gives the same case for determining the αk , and it is a simple matter to verify that each coefficient be in the same case regardless of whether the points are ω or 19 ω. Again, we begin by determining the appropriate equivalencies mod 3. Coefficient αA αB αC αD αA∪B αA∪C αA∪D Equivalence mod 3 -1 -1 0 -1 1 -1 1 π ∗ λ1 · C = 0 by Lemma 3.4. ⇒0= 2 (−(tA − 1)(tB + tC + tD − 1) − (tB − 1)(tA + tC + tD − 1) − tC (tA + tB + tD ) 27(T − 1) − (tD − 1)(tA + tB + tC − 1) + (tA + tB − 1)(tC + tD − 1) + (tA + tC − 1)(tB + tD − 1)) + (tA + tD − 1)(tB + tC − 1) Again, it is easy to verify this equality. A Case III ω e D ω e B C We first determine the equivalencies of the difference in ω, ω points for the relevant sets. Coefficient αA αB αC αD αA∪B αA∪C αA∪D Equivalence mod 3 -1 1 0 0 0 -1 -1 π∗ (λ1 ) · γ = 0 by Lemma 3.4. ⇒0= 2 (−(tA − 1)(tB + tC + tD − 1) − (tB − 1)(tA + tC + tD − 1) − tC (tA + tB + tD ) 27(T − 1) − tD (tA + tB + tC ) + (tA + tB )(tC + tD ) + (tA + tC − 1)(tB + tD − 1) + (tA + tD − 1)(tB + tC − 1)) 20 A Case IV D e e e e B C Again, this equality is easily verifiable. Similar to above, we first determine the equivalencies mod 3 and then verify the formula. Again, π∗ λ1 .C = 0 by Lemma 3.4. In fact, since every node has monodromy e, every coefficient is in the case of equivalence to 0 mod 3. ⇒0= 2 (−tA (tB + tC + tD ) − tB (tA + tC + tD ) − tC (tA + tB + tD ) 27(T − 1) − tD (tA + tB + tC ) + (tA + tB )(tC + tD ) + (tA + tC )(tB + tD ) + (tA + tD )(tB + tC )) (4) Again, it is a simple matter to check that this equality holds. Therefore, any one-dimensional boundary stratum in M0,T intersects with π∗ λ1 as predicted by our formula for the αji ’s. Finally, in order to get λ1 instead of its pushforward, we must pull the whole expression back by π. Since π has degree 31 , by Remark 3.2, 1 π ∗ π∗ (λ1 ) = λ1 3 ⇒ λ1 = 3π ∗ π∗ (λ1 ) X j j = 3π ∗ ( αi Di ) 21 4 The Second Chern Class of the Hodge Bundle on Spaces of Degree Two Admissible Covers. In this section we study the second Chern class of the Hodge bundle over spaces of degree two cyclic admissible covers. We first set up notation. In this section, let let Admg denote the space of admissible degree two covers of genus g. In some places it will be more efficient to identify the space of admissible covers by number of branch points and not genus. We will call the space of admissible covers with j branch points Admj . Finally, we will also work with spaces of admissible covers with i marked branch points and 1 marked identity point, we denote this space Admi,1 Our notation of boundary divisors in the space of admissible covers will vary as well in places. We let ∆i denote the boundary divisor with i branch points on the left component. Some proofs necessitate that we denote a divisor by its number of branch points on the left component and by the space it resides in. In these cases we let ∆ni denote that the divisor is in the space of admissible covers with n marked points. If an identity point is also present, the divisor will be denoted ∆i,1 Lemma 4.1. [1] Let λ1 be the first Chern class of the Hodge bundle over Admg . Furthermore, let ∆i denote the codimenesion 1 symmeterized stratum parameterizing nodal covers with i points on the left twig. Let N denote the total number of branch points on each divisor. Then, λ1 can be expressed as P αi ∆i over all symmetrized boundary divisors ∆i of Admg     i(N −i)   8(N i ≡ 0 mod 2 −1) where αi =    (i−1)(N −i−1)  i ≡ 1 mod 2  8(N −1) (5) Our main goal in this section is to extend these results to the second Chern class. The first step is a consequence of a theorem of Keel which states that λ2 can be expressed as a linear combination 22 of codim 2 boundary strata in Admg . Lemma 4.2. [3] Let λ2 denote the second Chern class of the Hodge bundle over Admg . Let ∆i1 i2 i3 denote the symmeterized stratum with i1 points on the left component, i2 points in the center, and i3 components on the right. Then, there exists rational coefficients αi1 ,i2 ,i3 such that λ2 = X αi1 ,i2 ,i3 ∆i1 i2 i3 Our theorem in this section determines these coefficients. To this end, we will use the following facts. Lemma 4.3. 1 1X n λ2 = λ21 = αi λ1 ∆ni 2 2 Proof. The first equality follows from Mumford’s relations[5] the second follows from Lemma 4.1 Following notation from Section 2.6 we have Lemma 4.4. αni λ1 ∆ni = αni ([1]L [λ1 ]R ⊕ [1]R [λL 1 ]) =      αni ([1]|Admi,1 [λ1 ]|Admn−i,1 ⊕ [λ1 ]|Admi,1 [1]|Admn−i,1 ) i ≡ 0 mod 2     αni ([1]|Admi+1 [λ1 ]|Admn−i+1 ⊕ [λ1 ]|Admi+1 [1]|Admn−i+1 ) ≡ 1 mod 2 Lemma 4.5. [λ1 ]|Admn−i,1 = X [λ1 ]|Admi,1 = X [λ1 ]|Admn−i+1 = X [λ1 ]|Admi+1 = αij ∆ij,1 n−i+1 αn−i+1 j+1 ∆j+1 X 23 n−i αjn−i ∆j,1 i+1 αi+1 j+1 ∆j+1 Proof. For the case with with no identity point simply apply lemma 4.1. an identity point note that [λ1 ]|Admi,1 = P αij,1 ∆ij,1 . For the case with Now consider the forgetful morphism φ : Admi,1 → Admi . Then αij,1 ∆ij,1 = φ∗ (αj ∆j ) so that αj,1 = αj The strategy of the proof for the following theorem is to fix a divisor ∆i1 ,i2 ,i3 in the boundary expression for λ2 and track when it appears in the expression of lemma 4.5. This allows us write the coefficient αi1 ,i2 i3 in terms of the coefficients αi , the latter of which we have an explicit formula for. Theorem 4.1. Let ∆i1 ,i2 ,i3 denote the codimension 2 symmeterized stratum in Admg with i1 branch points on the left component, i2 branch points on the middle component, and i3 branch points on P the right component. Then λ2 = 2 αi1 ,i2 ,i3 ∆i1 ,i2 ,i3 where     i1 i2 i3 (2i1 i2 +2i1 i3 +2i2 i3 −i1 −2i2 −i3 )    32(i1 +i2 +i3 −1)(i1 +i2 −1)(i2 +i3 −1)     αi1 ,i2 ,i3 = (i1 −1)(i2 )(i3 −1)((i2 +i3 −1)(i1 +i2 )+(i1 +i2 −1)(i2 +i3 ))  32(i1 +i2 +i3 −1)(i1 +i2 )(i2 +i3 )        (i −1)(i2 +i3 −1)(i2 +1)(i3 )(i1 +i2 −1)+(i3 )(i1 +i2 )(i2 −1)(i1 −1)(i2 +i3 )   1 32(i1 +i2 +i3 −1)(i2 +i3 )(i1 +i2 −1) i1 , i2 , i3 ≡ 0 mod 2 i1 , i3 ≡ 1, i2 ≡ 0 mod 2 i1 , i2 ≡ 1, i3 ≡ 0 mod 2 Proof. Case 1: i1 , i2 , i3 even Let i1 + i2 + i3 = n. Since i1 , i2 , i3 are even, all nodes of ∆i1 ,i2 ,i3 are unramified. Thus the divisor ∆i1 ,i2 ,i3 only appears in the case i even of lemma 4.5. Using this, we have that X αi1 ,i2 ,i3 ∆i1 ,i2 ,i3 = X αni ([1]Admi,1 × X n−i αjn−i ∆j,1 ⊕ [1]Admn−i,1 × X αij ∆ij,1 ) (6) n−i Now we fix a given ∆i1 i2 i3 and compute its’ coefficient in (6). The expression [1]Admi,1 × ∆j,1 gives the divisor ∆i1 i2 i3 when i = i1 and j = i2 , or i = i3 and j = i2 . The expression [1]Admn−i,1 × ∆ij,1 gives the divisor ∆i1 i2 i3 when i = i2 + i3 and j = i2 , or when i = i1 + i2 and j = i2 . Combining the above lemmas we have the following for the coefficient of a fixed ∆i1 i2 i3 24 αni1 αii22 +i3 + αni3 αii21 +i2 + αni2 +i3 αii22 +i3 + αni1 +i2 αii21 +i2 = αi1 i2 i3 Using the expressions αni1 = αni2 +i3 and αni3 = αni1 +i2 we have the expression αni1 αii22 +i3 + αni3 αii21 +i2 = 1 αi i i 2 123 which holds when i1 , i2 , i3 are even. Plugging in the formula for each α from (4) into the above we i1 i2 1 +i2 )i3 get the coefficient 2( 8(i1(i+i + 2 +i3 −1) 8(i1 +i2 −1) i1 (i2 +i3 ) i2 i3 8(i1 +i2 +i3 −1) 8(i2 +i3 −1) ) for the divisor ∆i1 ,i2 ,i3 Upon simplification we arrive at the expression i1 i2 i3 (2i1 i2 + 2i1 i3 + 2i2 i3 − i1 − 2i2 − i3 ) 32(i1 + i2 + i3 − 1)(i1 + i2 − 1)(i2 + i3 − 1) Case 2: i1 , i3 odd, i2 even Since i1 , i3 are odd, all nodes of ∆i1 ,i2 ,i3 are ramified. Thus, the divisor ∆i1 ,i2 ,i3 only appears in the case i odd of lemma 4.5. Given this we have X αi1 ,i2 ,i3 ∆i1 ,i2 ,i3 = X αni ([1]Admi+1 × X n−i+1 ⊕ [1]Admn−i+1 × αn−i+1 j+1 ∆j+1 X i+1 αi+1 j+1 ∆j+1 ) (7) Again we fix a given ∆i1 i2 i3 and compute its’ coefficient in (7). The expression [1]Admi+1 × ∆n−i+1 j+1 gives the divisor ∆i1 ,i2 ,i3 when i = i1 and j = i2 or when i = i3 and j = i2 . The expression [1]Admn−i+1 × ∆i+1 j+1 gives the divisor ∆i1 ,i2 ,i3 when i = i2 + i3 and j = i2 + 1 or when i = i1 + i2 and j = i2 . Combining these gives the following coefficient for a fixed ∆i1 ,i2 ,i3 +i3 +1 +i2 +1 +i3 +1 +i2 +1 αni1 αii22 +1 + αni3 αii21 +1 + αni2 +i3 αii22 +1 + αni1 +i2 αii21 +1 = αi1 i2 i3 which can be regrouped into 1 +i2 +1 3 +1 αni1 αii22 +i + αni3 αii21 +1 = αi1 i2 i3 +1 2 Here, i1 is odd, i2 + 1 is odd and i3 is odd. Using this we plug in the expressions from Lemma 4.1. αi1 ,i2 ,i3 = 2( (i1 − 1)(n − i1 − 1) (i2 )(i3 − 1) (i3 − 1)(n − i3 − 1) (i2 )(i1 − 1) + ) 8(n − 1) 8(i2 + i3 ) 8(n − 1) 8(i1 + i2 ) 25 or αi1 ,i2 ,i3 = (i1 − 1)(i2 )(i3 − 1)((i2 + i3 − 1)(i1 + i2 ) + (i1 + i2 − 1)(i2 + i3 )) 32(i1 + i2 + i3 − 1)(i1 + i2 )(i2 + i3 ) Case 3: i1 odd i2 odd i3 even In this case the left node is ramified and the other is unramified. Thus the divisor ∆i1 ,i2 ,i3 appears in lemma 4.5 as follows. X αi1 ,i2 ,i3 ∆i1 ,i2 ,i3 = X αni ([1]Admi+1 × + X X n−i+1 αn−i+1 ⊕ [1]Admn−i+1 × j+1 ∆j+1 αnt ([1]Admn−t,1 × X X αtj ∆tj,1 ⊕ [1]Admn−t,1 × i+1 αi+1 j+1 ∆j+1 ) X αtj ∆tj,1 )) (8) Now we fix the divisor ∆i1 ,i2 ,i3 and see when it appears in (8). The divisor is given when i = i1 and j = i2 , n − i = i1 and j = i2 or when t = i1 + i2 and j = i2 , n − t = i3 and j = i2 . This gives the following equation for the fixed coefficient αi1 ,i2 ,i3 . +i3 +1 αi1 ,i2 ,i3 = 2(αni1 αii22 +1 + αni3 αii21 +i2 ) Here, i1 is odd, i2 + 1 is even, i3 is even and i2 is odd. Using this and the formulas from Lemma 4.1 we have αi1 ,i2 ,i3 = 2( (i1 − 1)(n − i1 − 1) (i2 + 1)(i3 ) (i3 )(n − i3 ) (i2 − 1)(i1 − 1) + ) 8(n − 1) 8(i2 + i3 ) 8(n − 1) 8(i1 + i2 − 1) or αi1 ,i2 ,i3 = 5 (i1 − 1)(i2 + i3 − 1)(i2 + 1)(i3 )(i1 + i2 − 1) + (i3 )(i1 + i2 )(i2 − 1)(i1 − 1)(i2 + i3 ) 32(i1 + i2 + i3 − 1)(i2 + i3 )(i1 + i2 − 1) Hodge Integrals In this section we use a technique that is similar to the one used in [6]. 26 Theorem 5.1. The family of Hodge integrals Z λn+m−3 1 Admg(n|m) R λn+m−3 can be computed using the recursive formula 1     Z Z m n X X 2(i + j − 1)(T − i − j − 1) n + m − 3 n m i+j−2 =3 λn+m−i−j−2 λ1 1 9(T − 1) i+j−2 i j Adm(i+1|j) Adm(n−i|m−j+1) i=0 j=0 Proof. From Theorem 1.1, we can replace one λ1 with its boundary expression, so the Hodge integral becomes Z λn+m−3 1 =3 Admg(n|m) n X m X i=0 j=0 Now for a given Dij we must evaluate R αji Z λ1n+m−4 π ∗ (Dji ) Admg(n|m) n+m−4 ∗ π (Dji ). g(n|m) λ1 (9) When we restrict λ1 to the pull back j j R of Dji we use the Whitney formulas, λ1 |Dij = λL 1 |Di + λ1 |Di . We raise this to the n + m − 4 power using the Binomial expansion theorem and have λn+m−4 |Dji 1 = n+m−4 X  k=0  n+m−4 j n+m−4−k i k R (λL 1 |Dj ) (λ1 |Di ) k First, recall that in Lemma 3.2 we proved that if i− j is divisible by 3 the above expression is zero. Furthermore, through a simple dimension count, we can see that the only non-zero term in this sum be when k = i + j − 2. If we account for the fact that there are n
m
j i irreducible boundary divisors in a given Dji , then we can see that Z λn+m−4 π ∗ (Dji ) 1 Admg(n|m) =  n+m−3 i+j −2    Z Z n m i+j−2 λ1 λn+m−i−j−2 1 i j Admg(i+1|j) Admg(n−i|m−j+1) Placing this and the expression for αji when i − j is not divisible by 3 into (7) completes the proof. Remark 5.1. In order for the values of i, j to define stable admissible cover spaces in the terms in Theorem 5.1, they must satisfy the following properties: 1. i − j ≡ 2 mod 3 2. i + j ≥ 2 27 It is also easy to note that from the combinatorial factor, the term when i = n − 1, j = m must always equal zero. This, and the aforementioned conditions on i, j, greatly reduces the number of terms which may give a non-zero contribution to the sum given in Theorem 5.1. Also, notice we only allow i − j ≡ 2 mod 3. If we allowed i − j ≡ 1 mod 3, due to the symmetry of boundary divisors, we would get exactly the same sum and would thus need to multiply the sum by an extra factor of 12 . We use this formula to compute Z λ1n+m−3 Admg(n|m) for some values of n, m in the following table. R λn+m−3 1 n m 3 0 1 3 2 2 2 9 4 1 4 27 6 0 8 27 3 3 128 135 5 2 3392 729 4 4 446923 5103 Note that we do not include symmetric cases (where the values of n and m are switched) as the Hodge integrals are equal due to the symmetry of boundary divisors. References [1] José Bertin and Matthieu Romagny. Champs de Hurwitz. Mém. Soc. Math. Fr. (N.S.), (125126):219, 2011. 28 [2] Charles Cadman and Renzo Cavalieri. Gerby localization, Z3 -Hodge integrals and the GW theory of [C3 /Z3 ]. Amer. J. Math., 131(4):1009–1046, 2009. [3] Sean Keel. Intersection theory of moduli space of stable n-pointed curves of genus zero. Trans. Amer. Math. Soc., 330(2):545–574, 1992. [4] John W. Milnor and James D. Stasheff. Characteristic classes. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76. [5] David Mumford. Towards an enumerative geometry of the moduli space of curves. In Arithmetic and geometry, Vol. II, volume 36 of Progr. Math., pages 271–328. Birkhäuser Boston, Boston, MA, 1983. [6] Peter Troyan. Hyperelliptic hodge integrals, 2007. 29