THE PRODUCT RULE IN κ∗(Mct ) g,n I. SETAYESH 7 Abstract. We describe explicit formulas for the product rule in κ∗(Mct ). g,n 1 0 2 n a 1. Introduction J Let ǫ : M → M be the forgetful map, viewed as the universal curve over 8 g,n+1 g,n 1 M , the moduli space of stable curves of genus g with n marked points. Let L → g,n i M be the cotangent line bundle over M , whose fiber over a given curve is the ] g,n+1 g,n+1 G cotangent space at the ith marked point. Define A ψ = c (L ) ∈ A1(M ) and κ = ǫ (ψi+1) ∈ Ai(M ) . . i 1 i g,n+1 i ∗ n+1 g,n h t By restriction, one can define the psi and kappa classes over M , the moduli space a g,n m of smooth curves of genus g with n marked poines, and Mct , the moduli space of g,n [ curves of compact type. Consider the sub-ring of A∗(Mct ) generated by the kappa g,n 1 classes, and call it the kappa ring. In [3] Pandharipande described an additive basis v for the kappa ring. 3 9 Theorem. [3] Given D,n ∈ N, the set 9 4 {κ | p ∈ P(D,2g+n−D −2) } 0 p . 1 generates κD(Mct ) as a Q-vector space, and for n > 0 it is a Q-basis. g,n 0 7 The natural question to ask, as first raised by Pandharipande [3], is to determine 1 explicit formulas for the product rule in the kappa ring of Mct , that is the main result : g,n v of this paper. i X Theorem 1. Let A = {a ,··· ,a } be a multi-set of integers, and n,g,d ∈ N be r 1 k a integers such that d = 2g +n− a −2. In A∗(Mct ) we have: i g,n X κ ···κ = x κ a1 ak p p(A) p∈SP(A,d) X where ℓ(r) ℓ(p) (−1)ℓ(A)+ℓ(t)+ℓ(r)+M ℓ(t)−ℓ(r) x = (|r +1|)! (ℓ(r| )−1)! p (|t|+1)! M −ℓ(r) j pl t≤r≤p∈SP(A) (cid:18) (cid:19)j=1 l=1 X Y Y in which M = min{ℓ(t),d}, SP(A) (resp. SP(A,d)) is the set of partitions of the multi-set A (resp. into at most d parts) and ℓ(p) is the number of components of p (for notations see Definition 5 and Definition 9). 1 2 I. SETAYESH Plan of the paper. In Section 2 we review some known results relating kappa classes and pushforwards of the psi classes. Section 3 contains the proof of the main theorem. The main idea is to use the following theorem of Pandharipande. Theorem 2. [3] There is a canonical surjective map ι : κ∗(M ) → κ∗(Mct ) g,n 0,n+2g g,n which is an isomorphism for n > 0. This allows us to reduce the computation to the case of genus zero. In the genus zero case, by the work of Keel [2] we have a very good understanding of the Chow ring, and we can explicitly compute all the required classes. In Section 4 we prove the combinatorial identity used in the Section 3. Acknowledgement. I would like to thank M. Einollah Zade, M. Saghafian and E. Salavati for their collaboration in the proof of Theorem 22. I am also grateful to E. Eftekhary and F. Janda for helpful comments on an earlier version of this paper. This work was partially done when the author visited the Institute for Mathematical Research (FIM) in Zurich and the result was first presented in a lecture in the Ein- stein series in Algebraic Geometry at Humboldt University supported by the Einstein Stiftung in Berlin. 2. Kappa and Psi classes In this section we explain the relation between the kappa classes and pushforwards of the psi classes. Before stating the results, we need to fix some notations. Definition 3. Let πk : M → M denote the forgetful map which forgets the g,n g,n+k g,n last k marked points, and let q be the multi-set {q ,··· ,q }. We define: 1 k k • ψ(q) = ψ(q ,··· ,q ) := πk ψqi+1 . 1 k g,n ∗ n+i ! i=1 r (cid:0) (cid:1) Y • κ = κ := κ . q q1,···,qk qi i=1 Y Definition 4. Let A = {a ,··· ,a } be a multi-set. Given σ ∈ S with the cycle 1 k k decomposition σ = γ ...γ (including the 1-cycles). We define: 1 r • σ (a) := a . i j Xj∈γi • σ(A) = {σ (A),··· ,σ (A)} (as multi-set). 1 r Definition 5. Let A = {a ,··· ,a } be a multi-set, we denote the set of partitions of 1 k A by SP(A). Given p ∈ SP(A), we use the following notations. k • |A| = a . j j=1 X • SP(A;l) (resp. SP(A,l)) denotes the set of partitions of A with exactly (resp. at most) l parts. • We denote the components of p by p , i.e. p = p . i i G 3 • ℓ(p) denotes the number of components of p. Also we write ℓ(A) = k and from the context it should be clear whether we work with a multi-set or a partition, so there should be no confusions. • |p| = |p |,··· ,|p | (as multi-set) 1 ℓ(p) • By abuse of notation we write ψ(p) instead of ψ(|p|), i.e. (cid:8) (cid:9) ℓ(p) ψ(p) = ψ(|p|) = πℓ(p) ψ|pi|+1 . g,n ∗ n+i  i=1 (cid:0) (cid:1) Y   The relation between the kappa classes and pushforward of the psi classes, due to Faber, is (see [1]): ψ(a ,...,a ) = κ . 1 k σ(a) σX∈Sk Lemma 6. ([1], [3, Lemma 11]) The subset of Ad(M ) defined by g,n ψ(A) A ∈ P(d) and κ A ∈ P(d) A (cid:8) (cid:12) (cid:9) (cid:8) (cid:12) (cid:9) are related by an invertible l(cid:12)inear transformation ind(cid:12)ependent of g and n. The map in one direction is given by the Faber’s formula. The inverse map is given by the following proposition. Proposition 7. κ = (−1)n+ℓ(p)ψ(p). A p∈SP(A) X Proof. Given q ∈ SP(A), we say that a pair (p,σ) with p ∈ SP(A) and σ ∈ S ℓ(p) splits q if σ(|p|) = q. Using Faber’s formula we have: (−1)n+ℓ(p)ψ(p) = (−1)n+ℓ(p)κ σ(|p|) p∈XSP(A) p∈XSP(A)σ∈XSℓ(p) = (−1)n κ (−1)ℓ(p) . q   q∈SP(A) (p,σ)splitsq X X   n Let denote the number of partitions of a set of n elements into k parts (the k (cid:26) (cid:27) Stirling numbers of the second kind). Given a splitting (p,σ) of q = ∐r q , by i=1 i restriction of p to each q we obtain an element p of SP(q ) and the restriction of σ i i i is a permutation with only one cycle of length ℓ(p ). Using the identity i n n (−1)k(k −1)! = −δn k 1 k=1 (cid:26) (cid:27) X 4 I. SETAYESH for the Stirling numbers of the second kind (see [4]), we have: r (−1)ℓ(p) = (−1)ℓ(pi)(ℓ(p )−1)! i   (p,σ)spXlittingofq Yi=1 pi∈XSP(qi) r ℓ(qi) ℓ(q )  = (−1)k(k −1)! i  k  i=1 k=1 (cid:26) (cid:27) Y X r   = (−1)r δℓ(qi). 1 i=1 Y Hence if for some i we have ℓ(q ) > 1 then the coefficient of κ is zero. Therefore i q the only term with non-zero contribution is q = {a }∪{a }∪···∪{a }. Therefore 0 1 2 n we have: (−1)n+ℓ(p)ψ(p) = κ q 0 p∈SP(A) X = κ(A) (cid:3) Example 8. (−1)3+ℓ(p)ψ(p)= ψ(a,b,c)−ψ(a+b,c)−ψ(a+c,b)−ψ(b+c,a)+ψ(a+b+c) p∈SPX({a,b,c}) = (κ κ κ +κ κ +κ κ +κ κ +2κ ) a b c a+b c a+c b b+c a a+b+c −(κ κ +κ )−(κ κ +κ )−(κ κ +κ ) a+b c a+b+c a+c b a+b+c b+c a a+b+c +κ a+b+c = κ a+b+c 3. Product of kappa classes As we explained in the introduction, Theorem 2 shows that any relation in κ∗(M ) 0,n is also valid in κ∗(Mct ). Moreover for n > 1 all the relations in κ∗(Mct ) are obtained g,n g,n in this manner. Therefore in order to prove Theorem 1, it is enough to show that those relations hold in κ∗(M ). Thus we start by computing the product rule in the kappa 0,n ring of M , but we need to fix some more notations. 0,n Definition 9. Let A = {a ,··· ,a } be a multi-set. 1 k k • A! = a !. i i=1 • A+1Y= {a +1 | i = 1,··· ,k}. i |A| a +a +···+a 1 2 k • = . A a ,a ,··· ,a (cid:18) (cid:19) (cid:18) 1 2 k (cid:19) • Given p,q ∈ SP(A), we write q ≤ p if q is a refinement of p. We use the following notations. – [p : q] ∈ SP(|q|) denotes the partition induced from p on the multi-set |q|. Note that ℓ(p) = ℓ([p : q]). 5 – q| ∈ SP(p ) denotes the partition induced from q on the multi-set p pi i i (by restriction). |(|p|+1)| • λ = λ := (−1)n+ℓ(p) . A a1,···,an |p|+1 p∈SP(A) (cid:18) (cid:19) X In the following proposition we compute the product rule in the top degree of κ∗(M ). 0,n Proposition 10. Let A = {a ,··· ,a } be a multi-set of integers such that a = 1 k i i X n−3. Then in An−3(M ) we have: 0,n κ = λ κ . a1,···,al a1,···,al n−3 Proof. Since An−3(M ) = Q, we have to integrate both sides over M . Given 0,n 0,n p = ∐r p ∈ SP(A), we have: i=1 i r ψ(p) = ψ|pi|+1 n+i ZM0,n ZM0,n+r i=1 |p |+Y···+|p |+r 1 r = |p |+1,··· ,|p |+1 (cid:18) 1 r (cid:19) |(|p|+1)| = . |p|+1 (cid:18) (cid:19) (cid:3) Therefore the result follows from Proposition 7. Definition 11. Let A = {a ,··· ,a } be a multi-set, q ≤ p ∈ SP(A). 1 k ℓ(r)(|r +1|)! N = N := (−1)n+ℓ(r)(ℓ(r)−1)! j=1 j A a1,···,an (a +1)! Q i r∈SP(A) X ℓ(rQ) |r +1| = (−1)n+ℓ(r)(ℓ(r)−1)! j . r +1 r∈SP(A) j=1(cid:18) j (cid:19) X Y ℓ(p) N := N . p pi i=1 Y Note that N = 1. a1 Definition 12. A genus zero n−pointed stable weighted graph is a connected graph G together with a map m : V(G) → 2{1,···,n} that satisfies the following conditions. • The sets m(v) (for v ∈ V(G)) form a partition of {1,··· ,n}. We call m(v) the set of markings at the vertex v. • For each v ∈ V(G) we have d(v) := deg(v)+♯m(v) ≥ 3. • H1(G) = 0, i.e. G is a tree. 6 I. SETAYESH For any genus zero n−pointed stable weighted graph G, we have a cycle [G] ⊂ M 0,n obtained as follows. We take [G] to be the image of the map M → M 0,d(v) 0,n v∈V(G) Y obtained by gluing along the edges of G. Remark 13. In [2] Keel proved that A∗(M ) is generated by cycles of the form [G], 0,n and it has perfect pairing. Therefore given a Chow class α ∈ A∗(M ), in order to 0,n show that α vanishes it is enough to show that the pairing of α against all the classes of the form [G] are zero. Note that if α belongs to the kappa ring, then α.[G] depends only on the dimension of components of [G] and not the distribution of the markings. Thus α.[G] depends only on the dimension sequence of G, i.e. {d(v)−3 : v ∈ V(G)}. The following theorem describes an additive basis for the kappa ring of M . 0,n Theorem 14. [3] Given D ∈ N, a Q-basis of κD(M ) is given by 0,n {κ | p ∈ P(D,n−D −2) } . p Theorem 15. Let A = {a ,··· ,a } be a multi-set of integers, and n,d ∈ N be such 1 k that d = n− a −2. In A∗(M ) we have: i 0,n X κ =  λ N κ . a1,···,ak q [p:q] p p∈SP(A,d) q≤p X ℓ(Xq)≤d      Proof. Using Remark 13 it is enough to check that the integral of both sides against cycles of the form [G] are equal. Note that both sides belong to An−d−2(M ), so they 0,n can be paired with cycles with exactly (n−3)−(n−d−2) = d−1 nodes. Thus we have to consider cycles that their dimension sequence have exactly d terms. If the integral of the left (or right) hand side over [G] is non-zero, then there is a way to put κ ,··· ,κ on irreducible components of [G] such that the dimension a1 al of each component is equal to the sum of degree of kappa classes over it. Therefore there is a partition p of the multi-set {a ,··· ,a } (with at most d parts) such that the 1 l multi-set of dimension of irreducible components of [G] is equal to |p|. We call such G a p−graph, and call [G] a p−cycle. Therefore it is enough to check our claim for all the strata of the form [G], where G is a p−graph for some p ∈ SP(A,d). By Theorem 14, kappa classes of the from κ for p ∈ SP(A,d) form an additive p basis for the kappa ring, therefore for each p there exist (a unique) x such that p κ = x κ . (3.1) a1,···,al p p(A) p∈SP(A,d) X Hence we have to prove that x = λ N . (3.2) p q [p:q] q≤p X ℓ(q)≤d 7 We prove this by induction on ℓ(p). Given a partition p with ℓ(p) = d, we integrate both sides of (3.1) against a p−cycle [G]. By Proposition 10 the integral of the left side is λ , and the integral of the right hand side is x . Hence we get x = λ , which p p p p confirms 3.2. Fix p ∈ SP(A) with ℓ(p) = e < d. Similarly by Proposition 10 the integral of the left hand side of (3.1) is λ = λ , and the integral of the right hand side is given pi p by Y x λ . q [p:q] q≤p X ℓ(q)≤d Since any partition q in this sum, except p, has length at least e+1 by the induction hypothesis we know that x = λ N . q r [q:r] r≤q X ℓ(r)≤d Thus we obtain x = λ −  λ N λ p p r [q:r] [p:q]   Xq<p  Xr≤q  ℓ(q)≤d ℓ(r)≤d       = λ − λ N λ . p r [q:r] [p:q] ( ) r<p r≤q<p X X ℓ(r)≤d Let p = p ∐···∐p , in order to simplify the formulas we denote q| (resp. r| ) 1 e pi pi e by q′ (resp. r′), then λ = λ and N = N . Therefore we obtain: i i [p:q] |q′| [q:r] [q′:r′] i i i i=1 Y Y e x = λ − λ N λ p p r [q′:r′] |q′| i i i ( ) r<p r≤q<pi=1 X X Y ℓ(r)≤d e = λ − λ −N +  N λ . p r [p:r] [q′:r′] |q′| i i i ℓ(Xrr<)≤pd  Yi=1q′i∈rX′iS≤Pq(′ipi)  In the first line q < p is a propersub partition,and in the second identity we added the term −N and took the sum over all partitions. Therefore in order to finish the [p:r] proof, it is enough to check that the terms in the parenthesis are zero, which is the (cid:3) content of the following lemma. Lemma 16. Given a multi-set A and a partition s of A. We have N λ = 0. [p:s] |p| s≤p∈SP(A) X 8 I. SETAYESH Proof. We have N λ = N λ . [p:s] |p| p |p| s≤p∈SP(A) p∈SP(|s|) X X Hence it is enough to check that the term in the left is zero. We denote the multi-set s by B, and b := ♯B. ℓ(p) N λ = λ N p |p| |p| pi p∈SP(B) p∈SP(B) i=1 X X Y |q+1| = (−1)ℓ(p)+ℓ(q) · q+1 r≤p≤q∈SP(B) (cid:18) (cid:19) X ℓ(p)ℓ(r|pi) |rj +1| (−1)b+ℓ(r) i (ℓ(r| )−1)! rj +1 pi i=1 j=1 (cid:18) i (cid:19) Y Y ℓ(r|pi) (where r| = rj is the partition induced by r on the set p ) pi i i j=1 a The data of a triple r ≤ p ≤ q ∈ SP(B) is equivalent to the data of r ≤ q ∈ SP(B) plus the data of a partition p′ := [p : r] ≤ [q : r] ∈ SP(|r|). The data of a partition p′ := [p : r] ≤ [q : r] ∈ SP(|r|) is equivalent to the following. For each 1 ≤ l ≤ ℓ(q) = ℓ([q : r]), the data of a partition ℓ(p′) l p′ = p′j l l j=1 a of the multi-set [q : r] . l Note that any multi-set p induces a unique p′j and we have i l ℓ(r| ) = number of components or r in p pi i = number of elements of p′j as a partition of the multi-set |r| l = ℓ(p′j) l Also if we vary i and j, rj runs over all components of r, hence i ℓ(p)ℓ(r|pi) |rj +1| ℓ(r|) |r +1| i = i . rj +1 r +1 i=1 j=1 (cid:18) i (cid:19) i=1 (cid:18) i (cid:19) Y Y Y Therefore 9 ℓ(r|) |q+1| |r +1| N λ = (−1)ℓ(q)+b+ℓ(r) i · p |p| q+1 r +1 p∈SP(B) r≤q∈SP(B) (cid:18) (cid:19) i=1 (cid:18) i (cid:19) X X Y ℓ(q) ℓ(p′) l (−1)ℓ(p′l) (ℓ(p′j)−1)!  l  Yl=1 p′l∈SXP([q:r]l) Yj=1   ℓ(t) Note that for a partition t of a set X there are exactly (ℓ(t )−1)! permutations σ j j=1 Y of X such that the partition of X obtained from the cycle decomposition of σ is equal to t. Hence we have ℓ(t) (−1)ℓ(t) (ℓ(t )−1)! = (−1)|σ|. j taparXtitionofX Yj=1 σX∈S|X| Therefore if r 6= q then the term |q| |pl| (−1)|pl| (|pj|−1)!  l  Yl=1 pl isaparXtitionofs(rl) Yj=1   vanishes, and if r = q we get (−1)ℓ(q). So |q| |q+1| |q +1| N λ = (−1)ℓ(q)+b i p s(p) q+1 q +1 p q (cid:18) (cid:19)i=1(cid:18) i (cid:19) X X Y b n−1 |B +1| (By Theorem 22) = (−1)k+1 (−1)|B| k −1 B +1 " # k=1 (cid:18) (cid:19) (cid:18) (cid:19) X = 0. (cid:3) Theorem 17. LetA = {a ,··· ,a }beamulti-set ofintegerssuchthatd = n−|A|−2. 1 k In A∗(M ) we have: 0,n κ = x κ a1,···,al p p(A) p∈SP(A,d) X where ℓ(r) ℓ(p) (−1)ℓ(A)+ℓ(t)+ℓ(r) ℓ(t)−ℓ(r) x = (|r +1|)! (ℓ(r| )−1)!·(−1)M p (|t|+1)! j pl M −ℓ(r) t≤r≤p j=1 l=1 (cid:18) (cid:19) X Y Y and M = min{ℓ(t),d}. 10 I. SETAYESH Proof. We define C (A) as follows. k C (A) = λ N k q |q| ℓ(q)=k X k ℓ(r) |r +1| = λ (−1)k+ℓ(r) j (ℓ(r)−1)! qi  r +1  q∈SP(A;k) i=1 ! j=1(cid:18) j (cid:19) X Y Y r∈SP(|q|)   The data of q ∈ SP(A;k) and r ∈ SP(|q|) is equivalent to the data of q ≤ r ∈ SP(A) with q ∈ SP(A;k), and this is equivalent to the data of r ∈ SP(A) plus the following. For each 1 ≤ j ≤ ℓ(r) a partition ℓ(q′) j q′ := q| = q′i j rj j i=1 a of the multi-set r , such that ℓ(q′) = k. For such pairs (q,r) and (r, q′ ), we j j j j have X|r +1| |r |+d (cid:8) (cid:9) j j j = , r +1 |q′|+1 (cid:18) j (cid:19) (cid:18) j (cid:19) and as in the proof of Lemma 16 we have λ = λ . qi q′i j i i,j Y Y ℓ(r) |r |+d dj C (A) = (−1)k+ℓ(r)(ℓ(r)−1)! j j λ k  |q′|+1 q′ji r∈XSP(A) Yj=1 q′j∈SXP(rj;dj)(cid:18) j (cid:19)Yi=1 (d1,···,dℓ(r))   Pdj=k = (−1)k+ℓ(r)(ℓ(r)−1)!· r∈SP(A) X (d1,···,dℓ(r)) Pdj=k ℓ(r) |rj|+dj dj (−1)ℓ(q′ji)+ℓ(tij) ||tij|+1|  |q′|+1  |ti|+1  Yj=1q′j∈SXP(rj;dj)(cid:18) j (cid:19) Yi=1 (cid:18) j (cid:19)   ti∈SP(q′i)    j j    = (−1)k+ℓ(r)(ℓ(r)−1)!· r∈SP(A) X (d1,···,dℓ(r)) Pdj=k ℓ(r) |r |+d dj ||q˜i|+1|  (−1)ℓ(rj)+ℓ(tj) j j j  |q˜ |+1  |q˜i|+1  Yj=1 tj∈XSP(rj) (cid:18) j (cid:19) Yi=1(cid:18) j (cid:19)  q˜j∈SP(|tj|;dj)    