Universal relations on stable map spaces in genus zero

We introduce a factorization for the map between moduli spaces of stable maps which forgets one marked point. This leads to a study of universal relations in the cohomology of stable map spaces in genus zero.


Introduction
The moduli spaces of stable maps M 0,m (X, d) provide examples of Deligne-Mumford stacks whose intersection theory is both accessible and interesting, as indicated by the considerable success of Gromov-Witten theory in genus 0. When the target is a point, M 0,m is a smooth projective variety and its cohomology ring has been computed by Keel ([Ke]). Recent studies lead to a more comprehensive view of the cohomology and Chow groups of these moduli spaces for other targets: [BO], [GP], [O1], [C1], [C2], [MM1], [MM2].
Let d ∈ H 2 (X) be a curve class on a smooth projective variety X. The space M 0,0 (X, d) parametrizes maps from rational smooth or nodal curves into X with image class d, such that any contracted component contains at least 3 nodes. Over M 0,0 (X, d) there exists a tower of moduli spaces of stable maps with marked points and morphisms f : M 0,m+1 (X, d) → M 0,m (X, d) forgetting one marked point, such that M 0,m+1 (X, d) is the universal family over M 0,m (X, d).
In this paper we introduce a factorization of the forgetful map f , which gradually contracts part of the boundary. This allows a detailed study of the cohomology and Chow rings of M 0,m+1 (X, d) as algebras over the rings of M 0,m (X, d). We find a series of universal relations over families of stable maps. (Theorems 3.3 and 3.5 in text).
This paper is the third in a series dedicated to the intersection rings of these moduli spaces. Previously we have found presentations for the Chow rings M 0,m (P n , d) for all m > 0 ( [MM1], [MM2]). The case m = 0 seemed less accessible from our point of view, as in a sense M 0,m (P n , d) has more structure when m > 0. A good parallel is in the study of the intersection ring for complete flag varieties versus that of the Grassmannian. In fact, when d = 1, M 0,0 (P n , 1) = Grass(P 1 , P n ), while M 0,1 (P n , 1) is a flag variety. This suggested an indirect approach, understanding H * (M 0,0 (P n , d)) by studying the extension of algebras H * (M 0,0 (P n , d)) → H * (M 0,1 (P n , d)). We view this step as prototypical for studies of Chow quotients by SL 2 -action from an intersection-theoretical point of view. The description of H * (M 0,0 (P n , d)) is completed in [MM3].
The relation to Chow quotients in the sense of [Ka] is as follows. We may regard the coarse scheme M 0,0 (X, d) as the SL 2 -Chow quotient of a simpler compactification for the space of maps P 1 → X × P 1 with image class (d, 1): the space Σ d X of quasi-maps of X, also known as the "linear sigma model". This is true at least for X convex. Let α be the class of the generic SL 2orbit in Σ d X . The Chow quotient of Σ d X is naturally a subvariety of the Chow variety Chow(Σ d X , α). The graph space U := M 0,0 (X ×P 1 , (d, 1)) is then the "universal family" over M 0,0 (X, d) in the sense of the above, admitting a canonical map into Σ d X . Moreover, intermediate spaces between U and Σ d X , as described in [MM1], appear naturally when one considers the geometry of the SL 2 -action. Finally, if B be the Borel subgroup of SL 2 , then M 0,1 (X, d) is the quotient U /B. Our study may thus be viewed as a procedure for understanding the intersection theory of SL 2 -Chow quotients Y /SL 2 for a projective variety Y , under suitable assumptions on the stabilizers. The first step regards the relation between the cohomology of Y and that of the quotient U /B, as in [MM1]. The second step concerns the morphism U /B → Y /SL 2 , as in this paper. [MM4] considers other applications of this viewpoint.
Another motivation for our study comes from the more complex situation of higher genus curves. Let g > 1. Then by methods similar to those presented in section 1 of this paper, the morphism M g,1 (P n , d) → M g,0 (P n , d) × M g,0 M g,1 factors in a series of birational morphisms with rational exceptional fibers. One may extract, for example, the Picard group for the basis in terms of that of the target. The line of inquiry opened in this paper partially applies to the maps above. We note that some of the exceptional loci have codimension greater than 1, however, this situation is remedied when restricting over each of the strata in the natural stratification M g,0 = M γ . Here γ denote stable graphs and M γ are open sets of closed substrata M γ , representing specific splitting types of a genus g curve. The case g = 1 is special and requires separate treatment.
The computations in this paper rely on two technical tools introduced in [MM1], [MM2]. One tool is the intersection ring B * associated to a network of local regular embeddings, motivated by the structure of anétale cover of the target stack. In the case of the moduli space of stable maps, the main advantage of B * over the usual intersection ring lies in the overall simplification of intersection on the boundary. Indeed, there is a natural stratification of M 0,m (X, d), indexed by stable trees with degree and marking decorations. In contrast to H * (M 0,m (X, d)), B * allows the classes of each of these strata to be decomposed as a polynomial of divisor classes in a natural way.
The second tool is a set of stability conditions for maps to projective varieties which engender various spaces birational to M g,m (X, d), via morphisms that contract the boundary of M g,m (X, d). We call these the intermediate spaces of M g,m (X, d). The fibered product of the universal families over these new spaces with M g,m (X, d) interpolates between M g,m+1 (X, d) and M g,m (X, d).
The sequence of spaces birational to M 0,0 (P n , d) as above is also constructed by Parker ([Par]) via GIT quotients under the SL 2 -action. Indeed, he regards M 0,0 (P n , d) as a GIT quotient of the graph space M 0,0 (P n × P 1 , (d, 1)) and its intermediate spaces as GIT quotients of the intermediate spaces for M 0,0 (P n × P 1 , (d, 1)) constructed in [MM1]. The existence of this contraction of M 0,0 (P n , d) was proved by different methods in [CHS].

A factorization of the forgetful map
1.1. Let m, n, d be natural numbers. We recall the various compactifications of M 0,m (P n , d) constructed in [MM2]: Fix a rational number a > 0 and an m-tuple A = (a 1 , ..., a m ) ∈ Q m such that 0 ≤ a j ≤ 1 for all j = 1, ..., m and such that m i=1 a i + da > 2. Definition 1.1. An (A, a)-stable family of degree d nodal maps from rational curves with m marked sections to P n consists of the following data: where L is a line bundle on C of degree d on each fiber C s , and e : O n+1 → L is a morphism of fiber bundles (specified up to isomorphisms of the target) such that: (1) ω C|S ( m i=1 a i p i ) ⊗ L a is relatively ample over S, (2) G := coker e, restricted over each fiber C s , is a skyscraper sheaf supported only on smooth points of C s , and (3) for any p ∈ S and for any I ⊆ {1, ..., m} (possibly empty) such that p = p i for any i ∈ S the following holds i∈I a i + a dim G p ≤ 1. MM2]) The moduli problem of (A, a)-stable degree d nodal maps with m marked points into P n is finely represented by a smooth Deligne-Mumford stack M 0,A (P n , d, a). For any other pair (A ′ , a ′ ) as above such that a i ≥ a ′ i for all i = 1, ..., m and a ≥ a ′ , there is a natural birational morphism M 0,A (P n , d, a) → M 0,A ′ (P n , d, a ′ ), a weighted blow-up along regular local embeddings.
The space U k is defined as the fiber product The existence of natural morphisms U k ′ → U k for k ′ < k is implied by the universality property of fiber products. The morphism from M 0,1 (P n , d) to U k contracts the components of degree no larger than k in each fiber over M 0,0 (P n , d).
Pullback of M 0,0 (P n , d,t) to U k yields analogousétale covers t U k (t).
The morphisms f k k+1 : U k → U k+1 may be well understood at the level these covers. When d is odd, the existence of enough distinct sections makes the morphism f ⌊(d−1)/2⌋ (t) : U ⌊(d−1)/2⌋ (t) → M 0,0 (P n , d,t) locally trivial. When d is even, the same reasoning applies on the complement of M (d/2). Let now h ⊂ {1, ..., d} be a cardinal (d/2)-subset, let M hh be defined as above. Consider two copies M h and Mh of M 0,2 (P n , d/2) × P n M 0,1 (P n , d/2). There is a fiber square diagram where the two fiber products M h and Mh are glued along the canonical section There is a natural Z 2 -action on the two spaces at the left side of the diagram, and the quotients are on the right side. The same symmetry is preserved after M h and Mh are contracted to two P 1 -bundles over M hh by the successive blow-downs down to the ((d − 2)/2)-th step. (The precise moduli problems for these contraction are defined in section 2). Similarly, there is a sequence of morphisms

Notation . Let
where ǫ is a small positive rational number, and let A k be the m-tuple consisting of one copy of 1 and m − 1 copies of a k . The space U k m is the pullback to M 0,m (P n , d) of the universal family over M 0,A k (P n , a k ). Let 1 be the marked point of weight 1 on the generic curve over M 0,A k (P n , a k ). Thus and U k m are also moduli spaces in their own right. Indeed, such a space is obtained as pullback of the universal family over M 0,A ′ k (P n , a k ), where the m-tuple A ′ k consists of one copy of 1, some copies of a k and other copies of a k−1 , in the desired order. The universal family is in itself a moduli space of weighted pointed stable maps, where the generic point of its fiber is assigned weight 0. Finally consists of the irreducible component of C x containing p 1 . On rigid covers, the above morphism always has three disjoint sections: p 1 , the universal section, and at least one hyperplane section.
1.2. The case of a general target. Let X be a smooth complex projective variety, and d ∈ H 2 (X) a class curve. A factorization of the forgetful morphism M 0,m+1 (X, d) → M 0,m (X, d) may be induced from any projective embedding of X. From the point of view of the boundary strata however, it is more natural to consider the embedding of X in a product of projective spaces as follows. Consider L 1 , ..., L s ∈ Pic(X) very ample, such that their first Chern classes generate the algebraic part of H 2 (X, Z). We consider the embedding of X given by all L i . To any curve class d ∈ H 2 (X) we assign an s-tuple Let n i := h 0 (L i ) − 1. Definition 1.1 above may be extended to the case when the target is i P n i by considering an s-tuple of weights a = (a 1 , ..., a s ), an s-tuple L : are still smooth Deligne-Mumford stacks, by the same reasons as for target P n .

The extended cohomology and Chow rings
The factorization of the forgetful map facilitates understanding the structure of the ring H * (M 0,m+1 (P n , d)) as an algebra over H * (M 0,m (P n , d)). Boundary classes are adjoined at each blow-up step, and codimension two relations among them exist. Additionally, U ⌊(d−1)/2⌋ → M 0,0 (P n , d) admits a relative cotangent class ψ', whose pullback is related to the canonical class ψ on M 0,1 (P n , d). The relation between intersection rings becomes truly transparent when we regard our spaces together with the networks generated by their canonical closed strata. Then relations among cohomology classes may be deduced by simple induction. Consider a general smooth projective target X ֒→ s i=1 P n i as in the preceding subsection, and d = (d 1 , ..., d s ). We denote by G = S d 1 × ... × S ds the product of s groups of permutations S d i . We recall succinctly the construction of a G-network of morphisms associated to U k m (X), and its extended cohomology ring. (see [MM1] and [MM2] for a more detailed motivation). We keep notations from the preceding section. Here we work in cohomology although the same construction works for Chow rings.
Let I be a set whose elements are of the form Definition 2.1. The space U k m,I (X) parametrizes degree d stable maps ϕ : (C, {p j } j=1,...,m ) → X together with a smooth point p m+1 ∈ C and marked closed curves {C h } h∈I such that: (1) p m+1 ∈ C ′ for any k-unstable curve C ′ ⊂ C.
(3) the incidence relations among the elements of I translate into analogous incidence relations among the curves C h : The space U has exactly three elements. Then M k 0,I parametrizes stable maps ϕ : C → X, together with one point p 1 ∈ C and splittings C h Ch = C for all (h,h) ∈ I satisfying all the relevant properties from Definition 2.1.
Let I be the set of all sets I as above. G acts on I by permutations. For each I ∈ I, let G I ⊂ G be the subgroup which fixes all elements of I. Any g ∈ G induces a canonical isomorphism g : U k m,I → U k m,g(I) , such that the following diagram is commutative whenever I ⊂ J.
In this paper all cohomology is considered with rational coefficients. The extended cohomology ring B * (U k m ) is constructed as follows: where the extended cohomology groups are The extended rings B * satisfy all the good properties of the usual intersection rings, under appropriate conditions. Thus pullback is well defined for morphisms compatible with the G-network. An example is provided by For each I as in Definition 2.1, consider the fiber square is constructed by concatenating all pullbacks f k ′ * k,m,I . These are obviously compatible with the equivalence relation of Definition 2.2.
When the target X is convex, all the moduli spaces U Another important feature is that the product D 2 h has a well known geometric significance. Indeed, the stratum of class D h is a fiber product of the form U 0 m,h = M h × X Mh, where M h and are themselves moduli spaces, Mh, and the product is along evaluation morphisms ev h : M h → X, evh : Mh → X There are canonical ψ-classes associated to ev h and evh. Their pullbacks ψ h and ψh are known to satisfy the relation .

Universal relations in cohomology and Chow rings
We will denote by D h the fundamental classes of boundary divisors in Consider the forgetful map f : M 0,m+1 (P n , d)) → M 0,m (P n , d)) with m canonical sections s i , and the evaluation maps ev i : M 0,m+1 (P n , d)) → P n , for i = 1, ..., m + 1. Let ψ i := c 1 (L i ), where the tautological line bundle L i is pullback by s i of the relative cotangent bundle of f . For any class α ∈ H * (X), we define the kappa class k(α) := f * ev * α.
H will denote the hyperplane divisor on P n . We recall the following comparison formula on M 0,m+1 (P n , d) where D i,m+1 is the Cartier divisor associated to the canonical section σ i of f m+1 (see for example [W]).
Lemma 3.1. The following codimension two class is zero in the cohomology ring H * (M 0,1 (P n , d)): Note that although the relation is written in B * (M 0,1 (P n , d)), it is in fact symmetrical in {h}, and thus it is a relation in H * (M 0,1 (P n , d)).

Proof. The formula can be checked by increasing induction on d. When
describes the usual relation in the cohomology of the flag space F ( * , P 1 , P n ), over that of the Grassmannian G(P 1 , P n ).
Assuming the formula true for all degrees less than d, we can derive a codimension two relation on any divisor D h . Indeed, D h is the class of a normal stratum U h := M 0,1 (P n , |h|) × P n M 0,2 (P n , |h|), where the fiber product is via evaluation maps at the first marked points of M 0,1 (P n , |h|) and M 0,2 (P n , |h|). Let π h and πh be the projections from U h on the first and second factors; let f h : M 0,1 (P n , |h|) → M 0,0 (P n , |h|), fh : M 0,2 (P n , |h|) → M 0,1 (P n , |h|) be the morphisms forgetting the first marked points. The evaluation maps at these points are ev h and evh. The following ψ classes on U h will play a role in the computation: ψ h := π * h ψ 1 and ψ, pullback of the ψ class on M 0,1 (P n , d). We . All the following arguments take place on U h . By the additivity of kappa classes ( [KK], Lemma 3.3), The induction hypothesis gives a formula for k h (H 3 ) as a quadratic expression in k h (H 2 ), ψ h and divisors {D h ′ } h ′ ⊂h . Simultaneously, pullback via fh of the analogous relation on M 0,1 (P n , |h|) expresses kh(H 3 ) as a quadratic function of kh( Furthermore, ψ h and π * h f * h ψ 2 may be written as: [LP], [MM1] Lemma 3.21, and a comparison formula (3.1)). Summation of k h (H 3 ) and kh(H 3 ) yields k(H 3 ) as an expression in ψ, k h (H 2 ), kh(H 2 ), and boundary divisors. The classes k h (H 2 ) and kh(H 2 ) can be written in terms of k(H 2 ), ψ and boundary divisors via the divisorial relation in Lemma 2.2 of [Pan]: on M 0,1 (P n , d), Pullback by πh • fh of the equivalent relation on M 0,1 (P n , |h|) is and thus, after eliminating H: Substituting the expressions for k h (H 2 ) and kh(H 2 ) into the above mentioned expression for k(H 3 ), we obtain the desired formula modulo the codimension two annihilator of D h in B * (M 0,1 (P n , d)). By Theorem 3.23 in [MM1], this annihilator is generated by ψDh, Thus in B * (M 0,1 (P n , d)), R = a(h), where a(h) is a linear combination of the terms above. Of necessity, a(h) = a(h ′ ) for any h, h ′ ⊂ {1, ..., d}. But there are no codimension two elements that annihilate all boundary divisors in B * (M 0,1 (P n , d)). Indeed, the only codimension two relations in B * (M 0,1 (P n , d)) are linear combinations of monomials The only common annihilators for these are of the form D h ′′ Dh′′, where h ′′ ⊃h or h ′′ ⊃h ′ . But these do not annihilate D h ′′ .
Notation . Let M hh := M 0,1 (P n , |h|)× P n M 0,1 (P n , |h|). Consider the gluing map M hh → M 0,0 (P n , d) and its class D hh in B * (M 0,0 (P n , d)). Let π h and πh denote the projections from M hh on the two components and let ψ h := π * h ψ 1 . The image of the class ψ h ∈ H * (M hh ) in B * (M 0,0 (P n , d)) is a degree 2 class denoted by F h .
Based on the structure of B * (M 0,1 (P n , k)) found in [MM1] Theorem 3.23, it is not hard to see that B * (M 0,0 (P n , d)) is generated over H * (M 0,0 (P n , d)) by the set of classes F h , D hh for h ⊂ {1, ..., d}. (In [MM1] we have worked with Chow rings; the entire argument works identically for cohomology).
Notation . Choose I := {h ⊂ {1, ..., d}, |h| > d/2} if d is odd, and let I additionally contain half of the sets h with |h| = d/2 if d is even, under the condition that no two sets h,h are simultaneously in I. We define the following classes in B * (M 0,1 (P n , d)): Dh′ for any h ∈ I. The class ψ I (h) is defined as ψ I (h) := ψ ′ I + D I (h). When d is odd, ψ ′ I is the pullback of the relative cotangent class for the morphism U ⌊(d−1)/2⌋ → M 0,0 (P n , d).
Remark 3.2. The following relation between ψ classes on M 0,m (P n , d) by Y.P. Lee and R. Pandharipande ([LP], Theorem 1) is instrumental in our computations ψ i + ψ j = D(i|j). Here m ≥ 2, i and j ≤ m, and D(i|j) is the divisor representing split curves, such that the marked points i and j lie in different components. With the notations from the previous sections, we will employ an analogous relation existing on intermediate spaces U k m , where m > 0 and k = (k 1 , {a i } i=2,...,m ) consists of a positive integer k 1 < d + m − 1, and weights a i on the marked points. Indeed, the class ψ 1 on M m+1 (P n , d) descends to ψ ′ 1 on each U k m , while for any j = 2, ..., m + 1, the class ψ j − j∈h D h , descends to a class ψ ′ j on U k m , where the sum above is after all divisors D h contracted by the morphism M m+1 (P n , d) → U k m , i.e. for all h = h 1 ⊔ M h such that |h| k := |h| k 1 +ǫ + i∈M h a i ≤ 1. Thus on U k m the following relation holds where the sum is taken after all h with |h| k > 1.
Next we show how the algebra B * (M 0,1 (P n , d)) may be constructed starting from B * (M 0,0 (P n , d)), by adjoining a divisor for each intermediate space U k . Each divisor comes with a natural quadratic relation. Thus, relation (1) below is pulled back from U ⌊(d−1)/2⌋ , while relations (2) and (3) are pulled back from B * (U |h|−1 ). We keep notations from Lemma 3.1 throughout.
Theorem 3.3. The algebra B * (M 0,1 (P n , d)) over B * (M 0,0 (P n , d)) is generated by divisors ψ ′ I and {D h } h∈I . The ideal of relations is generated by the following (1) ψ ′2 for a choice of the set I as above.
Note: We consider by convention Dh h = D hh , such that both terms Dh h D hh and D hh Dh h appear in relation (1) above.
When d is odd, U ⌊(d−1)/2⌋ is a projective bundle over M 0,0 (P n , d). Its cohomology ring is thus generated over H * (M 0,0 (P n , d)) by the cotangent class ψ ′ , which satisfies a degree 2 relation over the base ring. Pullback of ψ ′ to M 0,1 (P n , d) is ψ + h∈I D h . Substituting this in the relation of Lemma 3.1 yields (1). When d is even, we refer to the description of U (P n , d/2) is a contraction of M 0,2 (P n , d/2) as described in subsection 1.2, such that the marked points both have weight 1, and the map to P n has weight 1 d/2+ǫ . The fiber product is along the evaluation maps at the first marked point. The ψh denote the ψ class of U After comparing formulas (3.4) and (3.5) the their analogues forh, we obtain the following relation in the algebra B * (U |h|−1 ) over B * (M 0,0 (P n , d)) This formula accounts for the different versions of relation (1) depending on the choice of the set I above. Indeed, choose I such that h ∈ I. Then formula (3.6) is equivalent to I\{h}∪{h} − ψ ′2 I , which is exactly the difference between relation (1) applied to I and to I \ {h} {h}. Here we assumed the compatibility condition D h D h ′h′ = 0 for all h, h ′ such that |h| = |h ′ | = d/2 (relation (3) in lemma). Note that in B * (M 0,0 (P n , d)), there is also a compatibility condition D hh D h ′h′ = 0 for h, h ′ as above. Equation (3.6) may be recast into relation (2) of the Lemma via formula (2.1). We note that the same equation may be obtained in part II of the proof by formally (at the level ofétale covers) decomposing U (d−2)/2 → M 0,0 (P n , d) into a blow-down along the section S above, and a P 1 -bundle. Moreover, the reasoning employed in part II of the proof also guarantees that there are no other relations besides (1), (2), and (3)   LetX → X be a blow-up along a regularly embedded locus Y . Following [Ke] (Theorem 2 in Appendix), H * (X) may be written explicitly as an algebra over H * (X), provided that the pullback morphism H * (X) → H * (Y ) is surjective. In the presence of compatible G-networks in X, Y andX, the analogous statement holds for extended cohomology rings ( [MM1]). Note that in the present case H We verify the surjectivity condition here.
By induction, the algebra B * (U k+1 ) has generators ψ ′ and {D h ′ } h ′ over B * (M 0,0 (P n , d)), for all h ′ ∈ I such that |h ′ | < d − k − 1. On the other hand, the algebra B * (S h ) ∼ = B * (M hh ) over B * (M 0,0 (P n , d)) is generated by the divisor class f k+1 * h ψ h . By formulas (3.7) and (3.8) By [Ke], the ideal of relations in B * (U k ) is made of two parts: We proceed to find a and b in terms of the generators of B * (U k+1 ).
Let ψ(k + 1) denote the first Chern class of the relative dualizing sheaf for U k+1 → M 0,0 (P n , d). Thus pullback of ψ(k + 1) to M 0,1 (P n , d) is hh is a fiber product U k+1 1 (P n , |h|) × P n M 0,1 (P n , |h|), where U k ′ 1 (P n , |h|) is a contraction of M 0,2 (P n , |h|) as described in 1.2, such that both marked points have weight 1 and the map to P n has weight 1 k+ǫ . The fiber product is taken along the first marked point. Let ψ h andψ on U k+1 hh be pullbacks of the two ψ-classes from U k 1 (P n , |h|). We note thatψ differs from j k+1 * hh ψ(k + 1) by the class of the section j * [S h ]. Thus by formula (3.3), which may be reformulated as hh . On the other hand, by comparison formula (3.1) −j * [S h ] = f k+1 * hh ψ h − ψ h . Putting these two equations together, we obtain The case m ≥ 1. We have shown in section 1 how the forgetful morphism f m+1 : M 0,m+1 (P n , d) → M 0,m (P n , d) factors out into a series of blow-downs and the projection of a P 1 -bundle over M 0,m (P n , d). The first marked point was chosen to play a special role in our construction. The analysis done in the proof of Theorem 3.3. carries out to this case, with the extra simplification provided by the existence of sections in the intermediate spaces. Indeed, the P 1 -bundle over M 0,m (P n , d) admits a canonical section σ 1 . This determines a Cartier divisor D 1,m+1 on the bundle, whose class generates the cohomology of the bundle as an algebra over H * (M 0,m (P n , d)). Note that the subscript {1, m + 1} is a distinct convention from the subscripts h employed elsewhere. The same notation will be used for the pull-back of the above divisor to M 0,m+1 (P n , d). The following well-known relation holds on M 0,m (P n , d) (see for example [Pan]). whose class may be computed as above. With the notations from section 1, the exceptional divisor D h is paired with a strict transform D h\{m+1} of the divisor with the same name by the relation f * m+1 D h\{m+1} = D h +D h\{m+1} . The following theorem follows by the same arguments as for Theorem 3.3. (1) D 2 1,m+1 + f * m+1 ψ 1 · D 1,m+1 ; The relations of Theorems 3.3 and 3.4 characterize in general the role of the forgetful map in the cohomology of M 0,m (X, d), for any smooth projective target X. Moreover, via a suitable embedding of X in a product of projective spaces, these relations are refined to hold for all h defined as in section 1.2. In this sense we call them universal. However, for an arbitrary convex target, the above theorems do not exhaust the list of generators for the algebras B * (M 0,m+1 (X, d)) over B * (M 0,m (X, d)). Let us consider for example the case m = 0. With the notations from Theorem 3.3, the issue is that in general, the morphism j k+1 * hh : B * (U k+1 (X)) → B * (M hh (X)) may not be surjective, for example, when the cohomology ring of X is not generated by divisors. A class γ h ∈ B * (M hh (X)) which is not in the image of j k+1 * hh contributes the classes f k * k+1,h α h and f k * k+1,h α h D h to B * (U k (X)), while classes of the type f k * k+1,h α h D a h with a ≥ 2 can be written in terms of the above via relation (2) in Theorem 3.3.
Here we present a simple application to the case X = P n , d = 2.
Thus H * (M 0,0 (P n , 2)) may be recovered as the subring of invariants of B * (M 0,1 (P n , 2)) under an action of Z 2 × Z 2 . Extracting the invariant relations is a fun exercise.
The relations in H * (M 0,0 (P n , 2)) found above can be thus written as Y n+1 = 0, where This is consistent with the ring structure computed in [BO].