Tautological rings of stable map spaces

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1 Tautological rings of stable map spaces Anca M. Mustaţǎ, Andrei Mustaţǎ Scool of Matematical Sciences, 153 Aras Na Laoi, University College Cork, Cork, Ireland Abstract We find a set of generators and relations for te system of extended tautological rings associated to te moduli spaces of stable maps in genus zero, admitting a simple geometrical interpretation. In particular, wen te target is P n, tese give a complete presentation for te coomology and Cow rings in te cases wit/witout marked points. Key words: coomology, tautological, moduli spaces, stable maps Introduction Let d H 2 (X) be a curve class on a smoot projective variety X. Te space M 0,0 (X, d) parametrizes maps from rational smoot or nodal connected curves into X wit image class d, suc tat any contracted component contains at least 3 nodes. Over M 0,0 (X, d) tere exists a tower of moduli spaces of stable maps wit marked points and morpisms M 0,m+1 (X, d) M 0,m (X, d) forgetting one marked point. Moreover, M 0,m+1 (X, d), togeter wit an evaluation map ev m+1 : M 0,m+1 (X, d) X and m natural sections σ i : M 0,m (X, d) M 0,m+1 (X, d), form te universal family over M 0,m (X, d). In tis paper we investigate te relation between te structure of te coomology ring of te variety X and tat of M 0,m (X, d), wic is less tan obvious in particular wen m = 0. Wen marked points exist, pullback by te natural evaluation maps ev i : M 0,m (X, d) X generate a set of classes on te moduli space. A system of tautological rings for {M 0,m (X, d)} m is constructed from tese classes by analogy wit te moduli space of curves. addresses: A.Mustata@ucc.ie (Anca M. Mustaţǎ,), Andrei.mustata@ucc.ie (Andrei Mustaţǎ). Preprint submitted to Elsevier 3 October 2007

2 Wen te target space is an SL n flag variety, te tautological rings coincide wit te coomology rings as sown in ([12]). We expect tat te same result olds for a larger class of varieties. Te Gromov-Witten invariants of X and teir gravitational descendants are intersection numbers in tese rings. Special tautological classes are te first Cern classes ψ i of te tautological line bundles over M 0,m (X, d). Anoter special set of tautological classes are te k classes k a (α) defined in [6] by te formula k a (α) = f (ψm+1ev a+1 m+1α), were f : M 0,m+1 (X, d) M 0,m (X, d) is te forgetful map above. We define a set of extended tautological rings, offering a more convenient encoding of te boundary, and for wic te original tautological rings are invariant subrings. Teorems 2.5 and 2.6 in te text identify a set of generators and relations for tese rings in terms of boundary strata and k classes k 1 (α i ) of te generators {α i } i for te ring H (X). Relations in H (X) induce relations in te tautological rings via k class decomposition (formula (5) in text and formulas (1) and (2) in Teorem 2.6). Natural universal relations exist on te boundary. In particular, wen X = P n, te above form a complete presentation of te extended coomology (and Cow) rings of M 0,0 (P n, d), wit/ witout marked points, of a simple geometrical interpretation. (Teorem 3.3 in text). In particular we prove te Conjecture 4.19 of Berend-O Halloran in [3] regarding te coomology ring of M 0,0 (P n, 3). Te degree 2 case was proven by Berend and O Halloran in [3]. In [10], we also recovered te degree 2 case by metods related to tose of te present article. Most of tis work was written during te autors stay at te Matematical Sciences Researc Institute in Berkeley, to wic we are very grateful for its ospitality. 1 Definition of Tautological rings For any smoot projective target X and curve class d H 2 (X; Z), te moduli spaces M 0,m (X, d) parametrising m pointed stable maps of class d into X are Deligne-Mumford stacks saring a couple of common features wit te moduli spaces of stable curves: (1) natural forgetful morpisms f : M 0,m+1 (X, d) M 0,m (X, d); (2) natural gluing maps j : M 0,A1 { }(X, d 1 ) X M 0,{ } A2 (X, d 2 ) M 0,m (X, d), were A 1 A2 = {1,..., m} and d 1 + d 2 = d. 2

3 Here M 0,A1 { }(X, d 1 ) X M 0,{ } A2 (X, d 2 ) is te fiber product along te natural evaluation morpisms ev { } : M 0,Ai { }(X, d i ) X, were i {1, 2}. Te map j is a regular local embedding, and its image parametrizes maps from curves split into a degree d 1 and a degree d 2 connected curves, wit te distribution of marked points on te two curves indicated by te sets A 1 and A 2. We say tat te domain of j is a closed boundary stratum of M 0,m (X, d) of codimension 1. Oter closed boundary strata may be defined by increasing induction on codimension as follows. If j τ : M τ M 0,A1 { }(X, d 1 ) is a boundary stratum of codimension k in M 0,A1 { }(X, d 1 ), ten te composition j (j τ id M 0,{ } A2 (X,d 2 ) ) : M τ X M 0,{ } A2 (X, d 2 ) M 0,m (X, d) gives a boundary stratum of codimension k + 1 in M 0,m (X, d). Te unique features of M 0,m (X, d) consist in te existence of m evaluation maps into X, and te existence of virtual fundamental classes [M 0,m (X, d)] vir in Cow groups and omology ([1]). Tese classes are compatible wit te natural morpisms above ([2]) and compensate for te fact tat M 0,m (X, d) in general is not a smoot stack of te expected dimension. Wit te virtual fundamental classes and te natural morpisms defined above we can construct a system of rings inside H (M 0,m (X, d); Q). Tese are called te tautological rings, and are defined by analogy wit te moduli space of curves. We will work in omology of M 0,m (X, d) via te omomorpism σ : H (M 0,m (X, d)) H (M 0,m (X, d)) σ(β) = β [M 0,m (X, d)] vir. Te classes σ(ev i α) and teir pusforwards via forgetful and gluing maps are te first examples of tautological classes. We note tat te forgetful maps admit natural flat pullback in omology. Te gluing maps j of te boundary strata admit a Gysin map, wic will be denoted by j. Intersection of tautological classes in H (M 0,m (X, d)) can be defined in a canonical way. Consider tree moduli spaces M 1, M 2, M related by natu- 3

4 ral morpisms f 1, f 2, and te fibre square diagram f M 1 M M 1 2 M 2 f 2 f 2 M 1 f 1 M and let f := f 1 f 2 = f 2 f 1. Here f 1 and f 2 are strata embeddings, forgetful maps or compositions of tese, or te identity. Similarly we define σ : H (M i ) H (M i ) for i = 1, 2. σ(β) = β [M i ] vir, Definition 1.1 Te intersection product of two classes f 1 (σ(β 1 )) and f 2 (σ(β 2 )) is defined as f 1 (σ(β 1 )) f 2 (σ(β 2 )) := f ( f 1 β 2 f 2 β 1 f 1 ([M 2 ] vir )) Note tat by te axioms of te virtual class, f 1 ([M 2 ] vir ) = f 2 ([M 1 ] vir ) = f ([M] vir ), so te product is well defined. Te product is clearly associative. We may tus define Definition 1.2 Te tautological system of rings of X is te set of smallest Q algebras inside te omology groups satisfying te following properties: R(M 0,m (X, d)) H (M 0,m (X, d)) (1) R(M 0,m (X, d)) contains all classes σ(ev i (α)), were α H (X), i = 1,..., m. (2) Te system is closed under pus-forward via all forgetting maps: f : R(M 0,m+1 (X, d)) R(M 0,m (X, d)). (3) Te system is closed under pus-forward via all gluing maps: j : R(M 0,A1 { }(X, d 1 )) Q R(M 0,{ } A2 (X, d 2 )) R(M 0,m (X, d)) were A 1 A2 = {1,..., m} and d 1 + d 2 = d. (4) Te system is closed under te product defined above. Wen X is convex, te moduli spaces are smoot and tus te omomorpism σ is an isomorpism. Ten R(M 0,m (X, d)) may be viewed naturally as 4

5 a Q subalgebra of te bivariant coomology H (M 0,m (X, d)) and its definition coincides wit te one introduced in [12]. In te case of SL N -flag varieties, it was sown in [12] tat te coomology rings are tautological, and isomorpic to te Cow rings. In general we may ask for wic set of varieties te same property is true. Wit te tecniques of [8] and [10], weter te coomology of stable map spaces is tautological depends essentially on te structure of te linear sigma model (quasi-map space) of X. We may expect te answer to be affirmative in oter cases like smoot toric varieties. 1.1 Te extended tautological rings Consider a smoot projective variety X and an embedding X s i=1 P n i suc tat te algebraic part of H 2 (X, Z) is Z s, generated by first Cern classes of te very ample line bundles L 1,..., L s giving te embedding above. Tus every curve class d on X is Poincaré dual to te first Cern class of a line bundle s i=1l d i i for nonnegative integers d i. We write d = (d 1,..., d s ) Z s. Let G = S d1... S ds denote te product of s groups of permutations S di. We recall succinctly te construction of a G network of gluing morpisms for M := M 0,m (X, d), and its extended omology groups (see [8] and [9] for a more detailed presentation). Te nodes of te G network are te boundary strata defined at te beginning of tis section, and tey are connected via natural gluing morpisms. However, we need a way of indexing tese boundary strata tat is sensitive to teir natural automorpisms. Tis indexing, as described below, will igligt an action of te group G on eac node of te network, wic makes all morpisms in te network G equivariant. Consider te case wen m > 0. Let I denote a set wose elements are of te form = 1... s 1 suc tat eac i {1,..., d i } and 1 {2,..., m}. Assume tat =, or for all, I. We call suc I a nested set. In te case wen m = 0, let I denote any set of 2 partitions = i{1,..., d i } suc tat for any pair (, ), (, ) I, exactly tree of te sets,,, are non-empty. Suc I will be called a nested set. Definition 1.3 a) Assume m > 0. Te space M I parametrizes degree d stable maps ϕ : (C, {p j } j=1,...,m ) X togeter wit a set of closed connected curves {C } I suc tat (1) I, p 1 C C; (2) te degree of te map ϕ C is ( 1,..., s ), were = 1... s 1 and i denotes te cardinality of te set i ; (3) te incidence relations among te elements of I translate into analogous 5

6 incidence relations among te curves C : C C iff and C C = iff =. I, i {2,..., m}, p i C iff i 1. b)wen m = 0, let M I represent stable maps ϕ : C X, togeter wit marked splittings C C = C for all (, ) I satisfying conditions (1) and (2) from point a) above. In particular, M = M. By a sligt abuse of notation, we will denote M I as M wen I = {} and as M wen I = {(, )}. For any two nested sets I J, tere is natural local regular embedding φ I J : M J M I. Te spaces M I wit te morpisms φ I J form a network, on wic te group G acts naturally. For eac set I as above, let G I G be te subgroup wic fixes all elements of I. Any g G induces a canonical isomorpism g : M I M g(i). Definition 1.4 Te extended omology groups are B l (M 0,m (X, d)) := I H l (M I )/, te sum taken after all subsets I as above. Te equivalence relation is generated by: φ I J (α) g (α) [g] G I /G J for any α H (M J ) and J I. Te images of tautological classes in H l (M I ) via te projection H l (M I ) B l (M 0,m (X, d)) define tautological classes in B l (M 0,m (X, d)). Intersection of tautological classes in te extended omology groups is defined by combining Definition 1.1 above wit Definition 3.11 in [8], to obtain α β := φi J(α) I φ J I J(β) c top (φ I J I J N M I J M ), (1) were α H (M I ) and β H (M J ) are two tautological classes, N M I J M denotes te normal bundle of M I J in M. Here φ I I J and φ J I J are Gysin omomorpisms and te product φ I I J(α) φ J I J(β) is given by Definition 1.1. We note tat if α and β are tautological classes in H (M I ) and H (M J ) respectively, ten φ I I J(α) and φ J I J(β) are also tautological classes in β H (M I J ), as te operations listed in Definition 1.1 and Definition 1.2 are preserved by te Gysin maps on boundary strata. By te proof of Definition 3.11 in [8], te product above is well defined. 6

7 A system of extended tautological rings is tus constructed inside B (M 0,m (X, d)): Definition 1.5 Te system of extended tautological rings of X is te set of smallest Q algebras inside te omology groups T (M 0,m (X, d)) B (M 0,m (X, d)) satisfying all te properties listed in Definition 1.1, wit te maps φ I J standing for gluing morpisms. In [8], [10] we give a more detailed account of te extended intersection teory associated to te networks of boundary maps in a moduli space. Tus T (M 0,m (X, d)) enjoys te salient properties of R(M 0,m (X, d)), like existence of pusforward and pullback for te forgetful maps and for te gluing morpisms. Te most important feature of T (M 0,m (X, d)) is tat it allows te fundamental classes of boundary strata to be decomposed as polynomials of boundary divisors. Tis greatly simplifies te task of listing ring generators. On te oter and, R(M 0,m (X, d)) is te invariant subalgebra of T (M 0,m (X, d)) wit respect to te natural action of G. In [9] we sow ow an additive basis for R(M 0,m (P n, d)) may be extracted from te ring structure of T (M 0,m (P n, d)) in te case m > 0. 2 Structure of tautological rings Te set of stable map spaces for a target X may be refined by considering a larger variety of stability conditions. Notation For any m 0, let A, B be two disjoint sets suc tat A B = {1,..., m} and let d A, d B, d be curve classes on X suc tat d A + d B = d. We denote by D(A, d A B, d B ), te divisor of M 0,m (X, d) representing split curves, wit a degree d A component containing te set A of marked points, and a degree d B component containing te set B of marked points. Assume 1 B. Wit te notations from Definition 1.3, te support of D(A, d A B, d B ) is te image of te morpism M M 0,m (X, d) if m > 0, or M M 0,0 (X, d), were = 1... s A is suc tat ( 1,..., s ) = d A. In [10] we constructed a sequence of Deligne-Mumford stacks U k m, and of morpisms M 0,m+1 (X, d) = U 0 m... U lm m M 0,m (X, d), suc tat eac U k m is birational to M 0,m+1 (X, d). Wen m = 0 we drop te subscript m. Denote te morpisms above and teir compositions by fk k : U k m U k m for any k k, and f k : U k m M 0,m (X, d). Eac step fk k+1 is a 7

8 blow-up along a regularly embedded codimension two locus, suc tat pullback to M 0,m+1 (X, d) of te exceptional divisor is one of te divisors D(A, d A B, d B ) in M 0,m+1 (X, d) defined as above, wit 1 B (if m > 0) and m + 1 A. In te case m = 0, recall te embedding X s i=1 P n i introduced at te beginning of section 1.1, suc tat d = (d 1,..., d s ) Z s. For every pair (d A, d B ) as above, let d A = (d A,1,..., d A,s ) and d B = (d B,1,..., d B,s ), wit d A,i +d B,i = d i. Te divisors D({1}, d A, d B ) on M 0,1 (X, d) for wic s i=1 d A,i < s i=1 d B,i are exceptional divisors in te sequence of contractions above. In general, f lm : U lm m M 0,m (X, d) is a P 1 bundle. Te only exception is wen m = 0 and tere exists a class d suc tat 2d = d. Ten te map f lm may be formally understood at te level of étale covers as te a blow-up of a P 1 bundle. ( see [10] for a complete argument). Te morpisms making up te factorization of te forgetful map are locally complete intersection morpisms, so tey admit pullback in omology. Tautological and extended tautological rings T (U k m) may be defined in a natural way for eac of te contractions U k m, via [U k m] vir := f k [M 0,m (X, d)] vir. We note tat te functoriality of te virtual fundamental class implies [U 0 m] vir = [M 0,m+1 (X, d)] vir, and, in conjunction wit Proposition 6.7, b) in [4], also implies [U k m] vir = f 0 k [M 0,m+1 (X, d)] vir. We can tus define tautological classes on U k m as pus-forwards of tautological classes from M 0,m+1 (X, d) via te map fk 0 : M 0,m+1 (X, d) U k m, wile te elements of te extended tautological ring additionally include pus-forwards of tautological classes from te boundary strata of M 0,m+1 (X, d) to te boundary strata of U k m. Te multiplication of tautological classes follows from formula (1) in conjunction wit Definition 1.1. As a motivation for te above definition of tautological classes on te spaces U k m, we note te existence of natural evaluation morpisms evi k for 0 i m + 1, making te following diagram commutative M 0,m+1 (X, d) ev i X fk 0 ev k U k i m. Projection formula ten implies f 0 k (ev i β [M 0,m+1 (X, d)] vir ) = ev k i β [U k m] vir. Tus te definition of tautological classes on U k m parallels te one on M 0,m+1 (X, d). Let m = 0. For eac step k, let (, ) be as in Definition 1.3, b) suc tat si=1 i < s i=1 i. To fix notation for strata in te extended ring of U k, 8

9 consider te following diagram consisting of two fiber squares U k 1 k 1 U j k 1,j k 1 U k 1 f k 1 k f k 1 k U k j k U k f k M j f k M 0,0 (X, d). Here vertical morpisms are regular local embeddings of boundary strata. Te generic fiber of f k k 1 is an irreducible curve, wile f consists of two k morpisms fk k 1 and f k 1, eac wit irreducible generic fiber. f k k admits a natural section s k : M U k. Indeed, by definition, U k is te midpoint in a sequence of morpisms M 0,2 (X, ) X M 0,1 (X, ) U k M 0,1 (X, ) X M 0,1 (X, ) = M, wose composition is (f 2, id M 0,1 (X, ) ). Te forgetful map f 2 : M 0,2 (X, ) M 0,1 (X, ) admits a natural section, as M 0,2 (X, ) is isomorpic to te universal family over M 0,1 (X, ). Tis section, paired wit id M 0,1 (X, ) and ten composed wit projection on U k, induces s k. Let S k denote te image of s k, wit regular embedding j : S k U k, and j k := j k j. Te space U k 1 is te blow-up of U k along S, k and U k 1 is its exceptional divisor, wit projection g : U k 1 S. k Let f k 1 = f k f k k 1. Te space U k 1 divisors U k 1 is te strict transform of U k and f k 1 and U k 1 intersect along S k = S k. := f k f k 1 k. Te two Wen k = 1 i.e. U k 1 = M 0,1 (X, d) we drop te superscript k 1, writing simply f, f, j, j for te appropriate morpisms. Te virtual fundamental class of U k in T (U k ) will be denoted by D, and is te pullback of te analogous class on M 0,0 (X, d). Te classes of U k 1 and in U k are D and D, respectively. U k 1 We will denote by A k te set of all indices as above, suc tat D T (U k 1 ) are exceptional divisors for fk k 1. Note tat all suc differ only by permutations g G, for G defined as in section 1, and A k D is te unique exceptional divisor in R(U k 1 ). Let A k be te set of all indices, suc tat D is an exceptional divisor for f k 1 k, and k k. Te structure of te forgetful morpism leads to a natural decomposition of classes in omology. 9

10 From now on, to simplify notations, β will denote a class in H (M 0,m (X, d)) as well as its image in R(M 0,m (X, d)). Te same convention olds for classes on intermediate moduli spaces. As a preliminary, we recall te following relation over M 0,1 (P n, d) (found in [10], te proof of Lemma 3.1 ) ψ 2 N ψ ψd +, N D D = 4 d 3 k(h3 ) 3 d 4 k(h2 ) 2 (2) were N = N = N ψ = 2 (6 4 d2 d ), 2 (6 2 d 2 d d 4 if =. 3 2 ) if, and d d 2 Wit te notations from above, let ψ(k) denote te relative cotangent class for te morpism f k, were 0 k l 0. In particular, ψ(0) = ψ, te ψ class of M 0,1 (P n, d). Te natural relations among relative cotangent classes sum up to fk k 1 ψ(k) = ψ(k 1) D, A k f 0 l 0 ψ(l 0 ) = ψ A l0 D. Tis last relation and formula (2) imply tat ψ(l 0 ) 2 is te pullback of a class from M 0,0 (X, d) ([10] Teorem 3.3, (1)). In te case d = 2d, te same result is true for an appropriate definition of ψ(l 0 ) (as employed in [10] Teorem 3.3). Proposition 2.1 Let X be a projective variety. For eac integer k suc tat 0 k l 0, fix an element in A k. Te following relation olds in R(M 0,1 (X, d)) β = 1 2 [ψf f β + f f (ψβ) k j f f j β]. PROOF. Te above formula is te decomposition of te class β due to successive blowups. We proceed by descending induction on k. Te induction ypotesis states tat β = 1 2 [ψ(k)f k f k (β)+f k f k (ψ(k)β) (j k f k f k j k β+jk f k f k jk β)], k >k 10

11 for every β R(U k ). Here for eac k suc tat k < k l 0, is a fixed element in A k. Consider te case wen te curve class d cannot be written as 2d. Wen k = l 0, te tautological ring of te P 1 bundle Ū l 0 M 0,0 (X, d) is generated as a module over R(M 0,0 (X, d)) by two classes [Ū l 0 ] vir and ψ(l 0 ) [Ū l 0 ] vir. Writing any class β R(U l 0 ) as β = ψ(l 0 )f l 0 a + f l 0 b, wit a, b R(M 0,0 (X, d)), and pusing forward to M 0,0 (X, d), one obtains a = 1f l 0 2 β. Indeed, f l 0 (ψ(l 0 ) f l0 a) = 2a, wile f l 0 (f l0 b) = 0. Multiplying te above expression for β R(U l 0 ) by ψ(l 0 ) and pusing forward to M 0,0 (X, d) ten yields b = 1f l 0 2 (ψ(l 0 )β), as ψ(l 0 ) 2 [Ū l 0 ] vir was previously mentioned to be te pullback of a class from M 0,0 (X, d). We assume te induction formula true at te step k. Let β R(U k 1 ). Recall te Cartesian diagram g S k U k 1 j k 1 U U k, for A k fixed. Let ξ denote te first Cern class of O k 1 U (1), as well as its image in R(Ū k 1 ), and let D be te class of te exceptional divisor in R(U k 1 ). By Teorem 6.2 (b) in [4], β fk k 1 fk k 1 β restricts to zero on Ū k 1 \ j k 1 (Ū k 1 ), and tus β = j k 1 k 1 f k 1 k j k c + f k 1 k fk k 1 β, were c R(Ū k 1 ). We prove tat c = gg j k 1 β satisfies te equation above. Indeed, pus-forward fk k 1 of te above expression leads to as y = f k 1 k 0 = fk k 1 jk 1 c = jk g c, f k k 1 y for any y R(U k ). Tus on te P 1 bundle g : U k 1 S, k c = ξg α + g γ, wit α, γ R(S k ) suc tat 0 = j k α. However, as by projection formula j k 1 (ξg α) = D j k 1 g α = D f k 1 k j k α = 0, only te term gγ of c is relevant for te expression of β R(U k 1 ). Furtermore, applying g j k 1 to tis last mentioned expression we obtain g j k 1 β = g j k 1 j k 1 g γ = g (c 1 (N k 1 k 1) U U g γ). 11

12 k 1 Here N k 1 k 1 U denotes te normal bundle of U U in U k 1, wose first Cern class is ξ, and terefore We ave tus proved γ = g (ξ g γ) = g j k 1 β. β = j k 1 g g j k 1 β + f k 1 k fk k 1 β. (3) Applying te induction ypotesis to fk k 1 β and substituting te result in equation (3), we obtain β = j k 1 g g j k 1 β 1 2 f k 1 k + 1 (fk k 1 j k 2 f k f k j k k >k [ψ(k)f k f k 1 (β) + f k f k (ψ(k)f k 1 β)]+ f k 1 k β + f k 1 k j k f k f k jk f k 1 k We note tat ψ(k)fk k 1 β = f k k 1 k 1 (fk ψ(k)β) and tus comparison of te relative dualizing seaves fk k 1 ψ(k) = ψ(k 1) A k D leads to k β). β = j k 1 g g j k [jk 1 β 1 2 [ψ(k 1)f k 1 f k 1 jk (fk k 1 j k 2 f k f k j k k >k f k 1 f k 1 (β) + f k 1 f k 1 f k 1 k (β) + f k 1 f k 1 (ψ(k 1)β)]+ β + f k 1 k j k 1 jk 1 (β)]+ j k f k f k jk We recall tat by Definition 1.4, A k D β = j k 1 jk 1 β. Furtermore, f k 1 j = j k 1 f k 1 + j k 1 f k 1 and terefore f k 1 k β). = f k 1 j f k 1 jk 1 f k 1 f k 1 β = j k 1 Also, f k 1 k jk 1 + f k k 1 jk 1 = j k f k 1 f k k 1 j k 1 jk 1 (f k 1 jk 1 β = β) + j k 1, wic implies f k 1 (f k 1 jk 1 β). = j k 1 f k 1 j f k 1 β = j k 1 j k 1 jk 1 f k 1 f k 1 f k 1 f k 1 jk 1 β = β + j k 1 Note tat in fact f k 1 f k 1 = gg. Adding up we obtain f k 1 f k 1 jk 1 β. 12

13 = j k 1 f k 1 f k 1 j k 1 jk 1 f k 1 (f k 1 jk 1 β) + j k 1 β + j k 1 jk 1 f k 1 f k 1 f k 1 f k 1 jk 1 β = β + 2j k 1 g g j k 1 β. Remark also tat for any A >k, j k f k 1 k β = f k 1 k jk 1 β. Inputing tese last two relations in te last formula for β above results in te induction formula at step k 1. Wen d = 2d, te first induction step follows from te quasi-splitting of U l 0 into a blow-up and a P 1 bundle (see [10] for a detailed description of U l 0 in tis case). By te same logic, keeping te same notations as above, in te case m 1 we obtain Proposition 2.2 Let X be a convex projective variety and m 2. Te following relation olds in R(M 0,m (X, d)) β = ψ m f mf m β + f mf m (β D 1,m ) + m f {m}f j β, were f m : M 0,m (X, d) M 0,m 1 (X, d) is te forgetful map. In te term m f {m} f j β above, te sum is taken after all suc tat m. Tus altoug written as a sum in T (M 0,m (X, d)), tis term is in R(M 0,m (X, d)), as it is invariant under te G action. D 1,m denotes te Cartier divisor corresponding to te natural section σ 1 : M 0,m 1 (X, d) M 0,m (X, d). It parametrizes maps from rational connected curves containing a degree zero component on wic te markings 1 and m, as well as exactly one node of te curve, lie. PROOF. As in Proposition 2.1, te above formula is te decomposition of te class β due to te successive blow-ups U k m U k+1 m, for 0 k < l m and to te initial P 1 bundle f lm : U lm m M 0,m 1 (X, d). Te divisor class D 1,m may be tougt of as te pullback of te generator for te algebra R(U lm m ) over R(M 0,m 1 (X, d)). We will denote tat generator also by D 1,m. Tus any class 13

14 β R(U lm m ) can be written as β = D 1,m f lm a + f lm b, wit a, b R(M 0,m 1 (X, d)). As before, te coefficients a = f lm β and b = f lm (β D 1,m ) + ψ 1 f lm β are found by pusing forward β and D 1,m β to M 0,m 1 (X, d) and applying te projection formula. We recall te comparison formula for ψ 1 classes ψ 1 = f mψ 1 + D 1,m on M 0,m (X, d), wic in particular yields D 2 1,m + f mψ 1 D 1,m = D 1,m ψ 1 = 0. Tis last equation leads to te formula for b above. Te formulas for a and b and te comparison of ψ 1 classes imply β = ψ 1 f lm f lm β + f lm f lm (β D 1,m ) for any class β R(U lm m ). Te proposition now follows by induction after te blow-up step. We recall tat te exceptional divisors for te intermediate blow-ups are D {m}, te class of te stratum j k 1 {m} : U k 1 {m} U k 1 m for some k l m, suc tat m. Te maps from connected rational curves represented by points in U k 1 {m} ave te markings 1 and m on two different components. In tis context, equation (3) becomes β = j k 1 {m} f k 1 {m} f k 1 {m} jk 1 {m} β + f k k 1 fk k 1 β, for any β R(U k 1 m ), were fk k 1 : U k 1 m U k m is te blow-down morpism, and f k 1 k 1 {m} denotes te map from U {m} to te boundary stratum j : M M 0,m 1 (X, d). Analogously to te proof of Proposition 2.1, te formula f k 1 {m} jk 1 {m} β + f k 1 jk 1 β = jf k 1 β allows us to rewrite te term j k 1 {m} f k 1 {m} f k 1 {m} jk 1 k 1 {m} β above in T (U m ) as j k 1 {m} f k 1 {m} f k 1 jk 1 β D g() {m} f k 1 β. [g] G/G Tus by decreasing induction on k we obtain te formula stated by Proposition 2.2, after applying te cange of variable formula for ψ classes from [7], Teorem 1 ψ 1 + ψ m = D(1 m) := D {m}. m 14

15 In particular we may apply te propositions above to te tautological kappa classes, wose definition we recall below, following [6]. Definition 2.3 Let f : M 0,m+1 (X, d) M 0,m (X, d) denote te forgetful morpism and let α H (X). Te kappa class k a (α) := f (ψ a+1 ev α [M 0,m+1 (X, d)] vir ). Assume X s i=1 P n i suc tat any curve class d H 2 (X) may be assigned a unique tuple (d 1,..., d s ), and let D := s i=1 {1,..., d i }. For any = 1... s 1 D {1,..., m}, let := ( 1,..., s ). Let denote te complement of and consider te morpisms M 0, 1 {, }(X, ) X M 0, 1 { }(X, ) f M 0, 1 { }(X, ) X M 0, 1 { }(X, ) M 0,m (X, d), were f is te map forgetting te marked point on te first component, and identity on te second component. We define te class k,a (α) := f (ψ a+1 ev α! ([M 0,l+2 (X, )] vir [M 0,m l+1 (X, )] vir )) in H (M 0,l+1 (X, ) X M 0,m l+1 (X, )) and in T (M 0,m (X, d)). Here : M 0,l+2 (X, ) X M 0,m l+1 (X, ) M 0,l+2 (X, ) M 0,m l+1 (X, ) is te natural pullback of te diagonal X X X. Wen a = 1 we simply write k(α) := k 1 (α), k (α) := k, 1 (α) We note tat te same notation k (α) stands for classes on different spaces, depending on te coices of l and m. To avoid confusions we will specify our coices wenever k (α) is introduced in a new context. Corollary 2.4 Let X be a smoot projective variety and α H (X). (i) Te following relation olds in R(M 0,1 (X, d)) ev 1α = ψ 1 k(α) k 0 (α) + k (α). Here k (α) := k (α) in T (M 0,1 (X, d)), and eac k (α) is te class on M 0,1 (X, ) X M 0,2 (X, ) introduced at te end of Definition 2.3. In oter words, k (α) is represented by te space of stable maps wose domain splits into one curve intersecting te class α and anoter curve containing te marked point. 15

16 (ii) Let m 2. Te following cange of variable olds in R(M 0,m (X, d)) ev i α ev j α = ψ j k(α) k (i j) (α ). Here k (i j) (α ) := k (α), were k (α) is te class on M 0,{i, } (X, ) X M 0,{j, } (X, ) defined above. In oter words, k (i j) (α ) is represented by te space of stable maps wose domain splits into one curve intersecting te class α, on wic te marking i lies, and anoter curve containing te marking j. PROOF. First we note tat altoug te formula for k (α) was written in T (M 0,1 (X, d)), tis class is invariant under te action of te group G and tus lives in R(M 0,1 (X, d)). We also note tat k (α) = k j f f j ev 1α, wen te sum is as in Proposition 2.1. Tus applying Proposition 2.1 to ev 1α we obtain ev 1α = 1 2 [ψ 1f k(α) + f k 0 (α) k (α)]. Equation (i) follows after observing tat f k 0 (α) = k 0 (α) ev 1α, wic is a special case of Lemma 5.6 in [6]. For every j, l {1,..., m} let D j,l denote te class of te locus of maps from curves aving a degree 0 rational component tat contains exactly one node of te curve and te markings l and j. Let f : M 0,m (X, d) M 0,2 (X, d) be te map forgetting all but te markings i and j. Equation (ii) on M 0,m (X, d) is te pullback of te identical equation on M 0,2 (X, d). Indeed, te cange in ψ classes f ψ j = ψ j l {i,j} is compensated by te cange in kappa classes D j,l f k (i j) (α ) = k (i j) (α ) l {i,j} D j,l k(α). On M 0,{j,i} (X, d) we apply Proposition 2.2 to ev j α and obtain ev j α = ψ j k(α) + ev i α + k (i j) (α ), because f j f j (ev j α D i,j ) = ev i α and j f {j} f j ev j α = k (i j) (α ), were f j : M 0,{j,i} (X, d) M 0,{i} (X, d) is te forgetful map. Tis is one way to prove 16

17 equation (ii). Alternatively, we may pullback relation (i) via te two forgetful morpisms M 0,2 (X, d) M 0,1 (X, d) and apply te cange of variable formula [7], Teorem 1 for te ψ classes. Let H be a very ample divisor on X, let d := d H. Note tat by Lemma 2.2 of [13], k 0 (α) = 2 d k(hα) + 1 d 2 k(α)k(h2 ) + 2 d 2 k (α). (4) Equation (ii) in codimension one, togeter wit te cange of variable in ψ classes, as been employed by Lee and Pandaripande in [7] for a reconstruction teorem in quantum coomology, wit an analogue in quantum K-teory. For codimension two classes in te two plane Grassmaniann, tis equation as also been proved previously in [11]. After reading Corollary 2.4. Oprea informed us tat e can also prove statement (ii) by oter metods, involving projection formula via te forgetful map. Not surprisingly, te ensuing partial reconstruction for Gromov-Witten invariants ([11] for G(2, n)) may be obtained directly via WDVV equations. Let D = i{1,..., d i }. We will employ te notations introduced in Definition 1.3. D will denote te class of M in T (M 0,m+1 (X, d)). Teorem 2.5 Let X be a smoot projective variety, d H 2 (X) a curve class and let α 1,..., α s be generators for te ring H (X). As an algebra over T (M 0,m (X, d)), te extended tautological ring T (M 0,m+1 (X, d)) is generated by divisor classes D 1,m+1 and {D } m+1, were D {2,..., m + 1} and {m + 1}. te image in T (M 0,m+1 (X, d)) of products Q := i k (α i ) a i T (M ) wit as above. Here a i 0. Note tat te first set of divisors may be furter restricted due to te divisorial relations pulled back from M 0,4 (see [5]). Tis teorem is complementary to [10], Teorem 3.4, were universal relations among te ring generators are given. In addition, specific relations in T (M 0,m+1 (X, d)) come directly from relations in H (X), induced by te kappa class decomposition 17

18 k(αβ) = k(α)ev 1β + k(β)ev 1α + ψ 1 k(α)k(β) 1 k (α)k (β). (5) and its analogues on te boundary. Equation (5) can be deduced from Corollary 2.4, (ii) for te marked points m+ 1 and 1, after multiplication by ev m+1β and pus-forward by te forgetful map f : M 0,m+1 (X, d) M 0,m (X, d). We also employed te comparison formula f ψ 1 = ψ 1 D 1,m+1 togeter wit te observation tat f (D 1,m+1 ev m+1β) = ev 1β. A similar relation on M 0,0 (X, d) may be found in te next teorem. We now proceed wit te proof of Teorem 2.5. PROOF. We will first prove a sligtly different statement: Claim 1 Let m > 0. Te algebra T (M 0,m+1 (X, d)) is generated over Q by D 1,i and ev i α for i = 1,..., m + 1, k(α), D and k (α) (were does not necessarily contain m + 1). Wen m = 0, D 1,i is replaced by ψ 1 in te above. For any j, l {1,..., m + 1}, D j,l is as defined in te proof of Corollary 2.4. Here α varies among all classes in H (X). Note tat by Lemma 2.2. of [13] and Teorem 1 of [7], all te ψ classes at te marked points as well as nodes derive from te generators listed in te Claim. So are te classes ev α on te boundary, were is a node, due to Corollary 2.4 (ii). Let f denote any of te forgetful maps between two of te moduli spaces or normal strata, let g be any local embedding of boundary into a moduli space or into a boundary stratum, and let denote te tautological product on any stratum or moduli space. A monomial γ in te tautological ring is obtained by any sequence of operations f, g,, out of an initial set of inputs {evj α} j, were j is eiter a marked point or a node. Te claim above follows independently of m, by induction on te number of operations f performed inside te monomial. Indeed, wen tis number is 0, te monomial is of te form γ = ev l α l D a D b ij i,j, l i,j in agreement wit te claim. Let L m+1 be te Q-vector space < l ev l α l D a i,j D b ij i,j k(α r ) k (α r ) > r r, in T (M 0,m+1 (X, d)). Note tat L m+1 is closed under g and. Te induction step consists in cecking tat f (L m+1 ) L m. Coose γ L m+1 supported on a boundary stratum M I. Notice tat on M I, k(α) = f k(α) and k (α) = 18

19 f k (α) if m + 1, wile k (α) = k(α)f D \{m+1} k \{m+1} (α) oterwise. After projection formula it remains to understand te case γ = l ev l α l D a i,j D b ij i,j. Also by projection formula, te indices l may be restricted to l = m + 1, or nodes on components in te fiber wic are contracted by f. We may also assume j = m + 1. Wit te notations following Proposition 2.2, f (βd b i,m+1) = ( 1) b 1 σ i (β)ψ b 1 i for any class β T (M 0,m+1 (X, d)) and any integer b 1. Assume γ 0. Ten for l as above, σ i ev l α l = ev i α l, wile σ i D = D or D \{m+1}, depending on weter i or not. As all of te above are in L m, it remains to study γ = l ev l α l D a. We prove f (γ) L m by induction on te degrees a. If all a 1, te statement follows from te additivity of k classes (Lemma 3.3 in [6]). Assume a 2 for some and assume tat m+1 (te oter case works similarly). As f D = D {m+1} + D, it follows tat D a = Da 1 f D D a 2 (D {m+1} D ). (6) Furtermore, te support of D {m+1} D is isomorpic to te stratum M in M 0,m (X, d). Te induced isomorpism on tautological rings identifies boundary wit boundary, ev m+1α wit te pull-back at te node ev α, and D wit ψ, te ψ class at te node of te generic curve represented by M. Tese are all classes in L m. Tus formula (6) provides te induction step tat concludes te proof of te claim. Te claim implies te proposition after distinguising all te generators of te ring f T (M m (X, d)) from tose of T (M m+1 (X, d)). Note tat k(α) = f k(α) and ev i α = f ev i α in T (M m+1 (X, d)), for i m + 1. In te case i = m + 1 we apply Corollary 2.4, (ii). Finally, we reduce te k-classes k(α) to tose of te ring generators α 0,..., α s via te decomposition formula (5). Recall te notations following Definition 1.3. Tus for D and its complement, te space M is defined as M 0,1 (X, ) X M 0,1 (X, ). Let π, 19

20 π denote te projections on te components and let ψ, ψ be te pullbacks of te ψ class via te two projections. Te sum ψ + ψ may be written as D 2 in T (M 0,0(X, d)), were D is te virtual fundamental class of M. We will denote by ev = ev te evaluation map at te node. Te next teorem will include relations among kappa classes on te codimension one and two boundary of te moduli spaces. We introduce te following notations Notation For any α, β, γ H (X), identified wit teir images in R(X), and for any as above, let k (α β) be te class in T (M ) represented by te space of stable maps wose domain splits into one curve C intersecting te class α non-trivially, and anoter curve C intersecting te class β nontrivially, suc tat C as degree and contains te markings 1. Let k (α β γ) be te class in T (M ) representing stable maps wose domain splits as C C C, suc tat C C and C C. C as degree and contains te markings 1. C as nontrivial intersection wit α, C wit β and C wit γ. Te notations k (α β), k (α β γ) will also be employed for te images of te above classes in T (M 0,0 (X, d)). Wenever one of α, β, γ is te unit element in H (X), its spot in te formulas above will be left empty. As an example, we illustrate te generic curves and teir incidence wit elements of H (X) corresponding to te following tree kappa classes. k (α β γ), k ( α β), k (α β). α γ β α β α β Teorem 2.6 (i) Let X be a smoot projective variety and d H 2 (X) a curve class. Let α 0,..., α s be generators for te ring H (X), suc tat α 0 is te class of a very ample Cartier divisor. Ten te ring T (M 0,0 (X, d)) is generated by te following classes 20

21 k(α 3 0), k(α i ) and k(α 0 α i ) for i = 1,..., s; te images in T (M 0,0 (X, d)) of products Q := i k (α i ) a i ψ a T (M ), were a i 0 are integers and a = 0 or 1. (ii) Eac relation P (α 0,..., α s ) = 0 in H (X) induces relations P 1, P 0, and {Q P } in te extended tautological ring, were Q are classes as above. Here P a := k a (P (α 0,..., α s )) are polynomials of te generators listed above, derived by successive applications of te formulas (1) k(αβ) = 1 2 [k(α)k 0(β) + k 0 (α)k(β) k (α β)]. Here k (α β) := k (α)k (β) on M 0,1 (X, ) X M 0,1 (X, ). k 0 (α) is defined by equation (4). On te boundary (2) k (αβ) = 1 2 [k ( α β) + k ( β α) + k (α β) + k (β α) k (α β)ψ k (β α)ψ k (α)k 0 (β) k (β)k 0 (α)]. Here te intersection products k (α β)ψ, k (β α)ψ are computed in T (M ), and ten te image in T (M 0,0 (X, d)) of te resulting classes is entered in formula (2). PROOF. By te previous claim, elements in te extended tautological ring of M 0,0 (X, d) come as products on various strata of classes evα and f (ev α D a ), wit a, a 0 (note tat tese include k (α)). If a 1, ten te last term is k(α) or a linear combination of classes k (α) on te boundary. If a 2 for some, ten by te same procedure as in te proof of te claim we obtain products of classes D, ψ, ev α, or k (α) on te boundary. Moreover, Corollary 2.4, (i) applied to M 0,1 (X, ) and M 0,1 (X, ), togeter wit equations ev α = ev α and k,0 (α) + k,0 (α) = k 0 (α) on M result in te following formulas for ev α and k,0 (α) ev α = 1 2 [k 0(α)D + ψ k (α) + ψ k (α) k ( α) k ( α)], k,0 (α) = 1 2 [k 0(α)D ψ k (α) + ψ k (α) + k ( α) k ( α)]. 21

22 Here, k ( α) = k (α)d, in accord wit te notations preceding te teorem. We note tat ψ + ψ = D and k (α) + k (α) = k(α) on M. We ave tus sown tat te tautological ring of M is generated by classes D, ψ, and k (α), plus generators supported on te boundary of M. Here D is identified to te first Cern class of te normal bundle N M M 0,0 (X,d). Wile multiplication wit D in T (M ) is in fact a product in α k (α) aα T (M ) do not decompose T (M 0,0 (X, d)), classes of te form ψ a in T (M 0,0 (X, d)). Te only extra simplification we may afford is to consider a 1, because oterwise we can rewrite ψ 2 via te analogue of equation (2) on M. Consider generators α 0,..., α s for te ring H (X), and let α H (X). Te second step is to decompose k(α) and k (α) into polynomials of {k(α i ), k 0 (α i ) and k (α i )} i wit coefficients in Q[{D, ψ } ]. Tis is done by relations (1)- (2). Relation (1) is derived by multiplying bot sides of equation (i), Corollary 2.4 by ev β, and pusing forward by te forgetful map M 0,1 (X, d) M 0,0 (X, d). Note tat f k 0 (α) = k 0 (α) ev 1(α) as a special case of [6], Lemma 5.6. To justify relation (2), we combine te analog of equation (5) on M k (αβ) = k (α)ev (β) + ev (α)k (β) + ψ k (α)k (β) k (α)k (β), wit te formula for ev (α) and ev (β) written above. Here it is useful to notice tat k (α)k ( β) = k (α β), wile k (α)k ( β) = k ( α β) + k (α)k (β) on M. Finally, we note tat in te polynomial ring over Q generated by te ring generators listed above, polynomials P a wit a > 0 are in te ideal generated by P 0, P 1, and te boundary {Q P, 1 }, due to equation (2). Remark 2.7 Te generators listed in te two previous teorems take an especially simple form wen te coomology of X is generated by divisor classes. Ten te ring T (M 0,0 (X, d)) is generated by k(α 3 0), k(α 0 α i ), and boundary classes D and F, were F is te image of ψ in T (M 0,0 (X, d)). Te algebra T (M 0,m+1 (X, d)) is generated over T (M 0,m (X, d)) by boundary classes D and D 1,m+1 (ψ 1 in te case m = 0). 22

23 3 Maps to P n In te case of maps to P n, it turns out tat te tautological relations described in Teorem 2.6, togeter wit a set of natural universal relations on te boundary, completely determine te tautological ring structure. We recall tat in tis case te moduli space is a smoot Deligne-Mumford stack and its entire coomology is tautological. Te universal relations mentioned above are described in te following lemma. Notation Let M := M 0,1 (P n, ) P n M 0,1 (P n, ). Consider te boundary map g : M M 0,0 (P n, d) and its class D in T (M 0,0 (P n, d)). Let π and π denote te projections on te two components and let ψ := πψ 1, ψ := π (ψ 1 ). Te images of te classes ψ and ψ in T (M 0,0 (P n, d)) will be denoted by F and F. Te intersection of boundary strata D D wose generic member is illustrated below, is non-empty in one of te following tree cases \ \ =,,. In tese figures te component of te curve marked by as degree. On M, te ψ classes are well known to satisfy te relation ψ + ψ = D. Wen =, let D( ) := ; D. Te generic curve represented by te restriction of D( ) := ; D to D D is illustrated below. Teorem 1, [7] translates in te following relation between ψ classes on te intersection D D : ψ + ψ = D( ). In addition, we will see in te following lemma ow equation (2) applied on M induces, after suitable simplifications, a quadratic equation in ψ 23

24 ψ 2 + b ψ + c = 0, (7) wit te coefficients b and c described below. Notation Let H be te yperplane divisor in P n. Define + b := N ψ D + N ψ D N ψ D, c := 3 d 4 k(h2 ) 2 4 d 3 k(h3 ) + ( ) ( ) N D2 + N D 2 + ( ) ( ) N D D 2 N D D + 2 were N ψ and N are as in equation (2). Te classes D and ψ on M, togeter wit te standard relations among tem described above, turn out to completely define te algebra T (M 0,0 (P n, d)) in rapport wit H (M 0,0 (P n, d)). Lemma 3.1 Te algebra T (M 0,0 (P n, d)) is generated over H (M 0,0 (P n, d)) by codimension one classes {D } D and codimension two classes {F } D, satisfying te following relations: (1) D D = 0 wenever or or ; (2) F + F = D 2 ; (3) F D +F D = D( )D D for any, D suc tat ; (4) F 2 + b F D + c D 2 = 0 (5) F F = [b + D( )]F D + c D D for any, D suc tat. N D D PROOF. By definition, T (M 0,0 (P n, d)) is te extended coomology ring of te S d network generated by te regular local embeddings g ( see [8]). Tus te generators of H (M ) for all {1,..., d} form a complete set of generators for T (M 0,0 (P n, d)) over H (M 0,0 (P n, d)). Let f : M 0,1 (P n, ) M 0,0 (P n, ), f : M 0,1 (P n, ) M 0,0 (P n, ) be te forgetful morpisms. Denote by H := ev H, were ev : M 0,1 (P n, ) P n is te usual evaluation map, and similarly for. By Teorem 3.23 in [8], te algebra generators of H (M 0,1 (P n, )) over Q are ψ, H and boundary divisors D for. Similarly, te generators 24

25 of H (M 0,1 (P n, )) are ψ, H and boundary divisors D for. By Lemma 2.2, [13], k (H 2 ) can replace H among te generators. Moreover, tese two divisors can also be written as linear combinations of k(h 2 ), ψ and boundary divisors, via Lemma 2.2, [13] applied to bot M 0,1 (P n, ) and M 0,1 (P n, ), and te relations H = H, k (H 2 ) + k (H 2 ) = k(h 2 ) and ψ + ψ = D on M. Tus k (H 2 ) = ψ 2 d D + d k(h2 )+ 2 D d 2 D d. Following te above analysis, eac H (M ) contributes exactly two new generators D := [M ] and F := ψ D to te algebra T (M 0,0 (P n, d)) over H (M 0,0 (P n, d)). A priory, one sould also consider classes ψ k D wit k > 1, but tese will be written in terms of te classes above, due to te existence of a quadratic relation in ψ on M. Indeed, summing equation (2) for M 0,1 (P n, ) and M 0,1 (P n, ) yields k(h 3 ) in terms of ψ, ψ, k (H 2 ), k (H 2 ) and boundary divisors k(h 3 ) = k (H 3 ) + k (H 3 ) = = 3 4 k (H 2 ) ψ N D D k (H 2 ) ψ N D D,, N D D + N D D, were N = N = N = 2 2 (3 2 ), 2 4 ( ) if, and if =. We ave just seen ow ψ, k (H 2 ), k (H 2 ) are all expressions of k(h 2 ), ψ and boundary divisors. After substituting tese expressions in te equation for k(h 3 ) above, relation (7) ensues on M. Equation (2) in tis lemma is a standard relation between ψ classes. Equation (3) of te lemma is a corollary of Teorem 1 in [7]. Relation (4) of te lemma follows from equation (7) and, in conjunction wit Teorem 1, [7], implies relation (5) in te lemma. We will keep te notations from section 2 trougout. Wen te target is P n, te factorization of te forgetful map introduced in section 2 is M 0,1 (P n, d) = U 0... U (d 1)/2 M 0,0 (P n, d), 25

26 suc tat D is te exceptional divisor of U 1 wenever (d 1)/2, wile te support of D is te strict transform of a section S U U. We will also denote by S te image of tis section in U k, wen k >. Te structure of T (M 0,1 (P n, d)) as an algebra over T (M 0,0 (P n, d)) is completely determined due to te sequence above: eac divisor D is anniilated by ker(t (U 1 ) T (S )), and satisfies a quadratic equation ([10], Teorem 3.3). Te following corollary lists low degree relations in T (M 0,1 (P n, d)). Tese ave been proved already in [8], Teorem However, our point ere is tat tese relations derive from Lemma 3.1. due to te structure of te extension T (M 0,0 (P n, d)) T (M 0,1 (P n, d)) described above. Indeed, Relation (1) is derived from Lemma 3.1 (1), wile relation (2) is derived from Lemma 3.1 (3). Corollary 3.2 Te following relations old in T (M 0,1 (P n, d)). (1) D D = 0 wenever, or. (2) D D (ψ + D ) for all, suc tat =. PROOF. Assume 0 < < d/2. If D D = 0, ten on any intermediate space U k wit k d/2, te subspace S k U k is defined as in section 2. In te following we will drop te superscript k wenever te underlying space is clearly determined from te context. Restriction to te section S yields s D = 0, were s denotes te class of S on U. Tus by te considerations above, D D = 0 and also D D = 0 on all spaces U k wit k. After restriction to S and reiteration of te argument, we obtain D D = 0, D D = 0, and tus also D D = 0 and D D = 0 on all spaces U k wit k. We recall tat D + D = D. We now prove relation (2) under te same assumption as above. Te oter cases work similarly. First we notice te following Claim 2 D (a + bs ) = 0 on U D (a bd ) = 0 on U 1 wenever 0 < < d/2. Indeed, te quadratic relation satisfied by te exceptional divisor D may be written as (D D )(s D ) = 0. Tis immediately implies te claim. Recall tat any ψ class on M 0,3 (P n, d) may be written in terms of boundary divisors. (Tis is, for example, a consequence of [7], Teorem 1). Let U,, be te normal stratum of U wic is birational to M 0,1 (P n, ) P n 26

27 M 0,3 (P n, ) P n M 0,1 (P n, ). Te following analogue of te above property of ψ classes olds ere D D [ψ( ) + s + s, < D ] = 0, (8) were ψ( ) is te first Cern class of te relative dualizing seaf for f : U M 0,0 (X, d). Terefore, its pullback to U,, differs from te usual ψ class by s + s. By te Claim above, tis equation implies D D (ψ( ) + s, < on U. Indeed, tis is a consequence of te facts tat ψ( ) = f ψ( ) + < D ) = 0 D and s = f s <, D on U, as s s on U for all. Relation (1) also intervened in te computation. Again by te Claim, te desired relation (2) is obtained. We note tat equation (8) is te pullback of relation (3) in Lemma 3.1. Indeed, for any integer k suc tat k < d/2, s = (f k ψ + ψ(k) + I, < D ) on U k, and an analogue relation exists for s two formulas leads to equation (8). ([10], Teorem 3.3). Adding up tese Notation Consider te vector V l wit entries in T (M 0,0 (P n, d)) V l := (k(h l+1 ), k(h l ), {k (H l )} I, {ψ k (H l )} I ) T. Here te superscript T stands for taking te transpose. Teorem 2.6 gives a formula for a transition matrix A wit entries in Q[k(H 2 ), k(h 3 ), {D, F } I ], suc tat V l+1 = AV l for all l 1. Indeed, specializing equation (1) of Teorem 2.6 for α = H l 1, β = H 2, and equation (2) for α = H l, β = H yields all te 27

28 elements of A. Equation (4) is necessary for te transition matrix calculation, togeter wit te formula for k (H 2 ) derived in te proof of Lemma 3.1. Te open stratum component A o of A will be of special interest. It is te transition matrix for te vector Vl o := (k(h l+1 ), k(h l )) T in H (M 0,0 (P n, d)). 1 A o d = k(h2 ) 1 d k(h3 ) 1 k(h 2 ) 2 d 2, (9) 1 0 wile V o 1 = k(h2 ) and Vl+1 o = (A o ) l V1 o for l 1. d Teorem 3.3 Te extended coomology ring T (M 0,0 (P n, d)) is te Q algebra generated by classes k(h 2 ), k(h 3 ), and {D, F } I. A complete set of relations consists of all relations in Lemma 3.1, togeter wit te vectorial relation V n+1 = 0. PROOF. Te ring generators of T (M 0,0 (P n, d)) ave already been determined in Teorem 2.6. Indeed, we recall tat H (P n ; Q) = Q[H]/(H n+1 ). Let f : M 0,1 (P n, d) M 0,0 (P n, d) denote te forgetful map. In [10], a presentation of T (M 0,1 (P n, d)) as an algebra over T (M 0,0 (P n, d)) is found via te intermediate extensions {T (U l )} 1 l<d/2. Anoter presentation for te ring T (M 0,1 (P n, d)) appears in [8]. We recall ere tat te ring generators of T (M 0,1 (P n, d)) are H, {D } {1,...,d}, and k(h 2 ) = f k(h 2 ) (or, alternatively, ψ). A complete set of relations consists of tose listed in Corollary 3.2, plus ev H n+1 = 0, P = 0 and D P = 0, were P, P are degree n polynomials of variables ev H, {D } {1,...,d}, and f k(h 2 ), bot involving te monomial f k(h 2 ) n. In fact, P is te polynomial relation of minimal degree aving a summand of te type c f k(h 2 ) l, were c and l are positive constants. On te oter and, f k(h n+1 ) satisfies te same property, as sown by formula (9). Tis forces te equality P = af k(h n+1 ) modulo relations (1) and (2) of Corollary 3.2; ere a is a constant. Similarly, modulo P, P is te minimal degree polynomial relation aving a summand of te type c D f k(h 2 ) l. By formula (9) for te open stratum of M, and due to te dependence of k (H 2 ) on k(h 2 ) expressed in te Proof of Lemma 3.1, we again obtain tat P = a f k (H n+1 ) modulo te low degree relations on te boundary mentioned above. Here a is a constant. Furtermore, by Corollary 2.4, (i) ev H n+1 = ψk(h n+1 ) k 0 (H n+1 ) + k (H n+1 ). 28