Moduli Operad over F 1

Transcription

1 Moduli Operad over F 1 Yuri I. Manin and Matilde Marcolli Abstract. In this paper, we answer a question raised in [25], sec. 4, by showing that the genus zero moduli operad {M 0,n+1} can be endowed with natural descent data that allow it to be considered as the lift to Spec Z of an operad over F 1. The relevant descent data are affine torifications in the sense of [22]. More generally, we do the same for the operads {T d,n+1 } (whose components were) introduced in [6]. Finally, we discuss from this perspective the genus zero boundary modular operad {M 0 g,n+1} whose components are, by definition, unions of those boundary strata in {M g,n+1} that parametrize curves whose normalized irreducible components are projective lines. This is the operad in the category of DM stacks, so that for its complete treatment it would be necessary to develop a formalism of stacky F 1 geometry compatible with torifications as descent data. Introduction and summary Of many recently suggested definitions of F 1 geometry, we work with the one developed in [22] that seems to be the minimal one. Namely, an F 1 scheme is represented by its lift to Spec Z and the relevant descent data which are essentially a representation of the lifted scheme as a disjoint union of locally closed tori. An additional condition is the existence of a compatible affine covering: see the Definition 1.1. Further details are provided in sec. 1. In sec. 2, for each d 1, we introduce the operads with components {T d,n+1 } from [6] and define torifications on these components. In the case d = 1, where T 1,n = M 0,n+1, there are at least two different sources of such torifications; at present we do not know whether they determine equivalent F 1 descent data. We compute also a generating series for the numbers of F 1 m points of M 0,n, using Poincaré polynomials of T d,n computed in [6]. Finally, we make explicit a blueprint structure of M 0,n based upon explicit equations for M 0,n, as in [13], [18]. We also describe a blueprint structure on the genus zero boundary M 0 g,n+1 of the higher genus moduli spaces, using a crossed product construction. Sec. 3 is dedicated to the operadic structure morphisms and the problem of their compatibility with torifications. For d = 1, we can work with the usual modular operad of genus zero. Cyclicity fails for d > 1, however, for d = 1 the descent data to F 1 are compatible with the extra morphisms generating the cyclic operad structure. The second author acknowledges support and hospitality of the Max Planck Institute and the Mathematical Sciences Research Institute and support from NSF grants DMS , DMS , DMS , PHY

2 2 Yuri I. Manin and Matilde Marcolli 1. Reminder on torifications In the following, all our schemes and morphisms are defined over Z. Definition 1.1. ([22]). (i) A torification of the scheme X is a morphism of schemes e X : T X from a disjoint union of tori T = j I T i, T j = G dj m, such that the restriction of e X to each torus is an immersion (i. e. isomorphism with a locally closed subscheme), and e X induces bijections of k points, e X (k) : T (k) X(k), for every field k. (ii) The torification e X is called affine if there exists an affine covering {U α } of X compatible with e X in the following sense: for each affine open set U α in the covering, there is a subfamily of tori {T j j I α } in the torification e X such that the restriction of e X to the disjoint union of tori from this subfamily is a of torification of U α. (ii) A morphism of affinely torified schemes (torified morphism) Φ : (X, e X : T X X) (Y, e Y : T Y Y ) is a triple Φ = (φ, ψ, {φ i }) where φ : X Y is a morphism of Z-schemes, ψ : I X I Y is a map of the indexing sets of the two torifications, and φ j : T X,j T Y,ψ(j) is a morphism of algebraic groups, such that φ e X TX,j = e Y TY,ψ(j) φ j. Remark. We do not use the notion of affinely torified morphism introduced in [22], p Example 1.2. Any toric variety has a natural torification by torus orbits. In [22], explicit affine torifications are constructed, and it is checked that toric morphisms are compatible with them. This shows that the Losev Manin operad {L 0,n } invoked in [25] has natural descent data to F 1 also in the sense of this paper. 2. Affine torifications of T d,n 2.1. The varieties T d,n. We recall the construction of [6] of a family of varieties T d,n whose points parameterize stable n pointed rooted trees of projective spaces P d. They generalize the moduli spaces M 0,n, with the latter given by T 1,n = M 0,n+1. These varieties are also closely related to the Fulton MacPherson compactifications X[n] of configuration spaces, in the sense that, for any choice of a smooth complete variety X of dimension d, one can realize T d,n in a natural way as a subscheme of X[n].

3 Moduli Operad over F n pointed rooted trees of projective spaces. A graph τ consists of the data (F τ, V τ, δ τ, j τ ): a set of flags (half-edges) F τ ; a set of vertices V τ ; boundary map τ : F τ V τ that associates to each flag its boundary vertex; and finally the involution j τ : F τ F τ, jτ 2 = 1 that registers the matching of half edges forming the edges of τ. We consider here only graphs whose geometric realizations are trees, i. e. they are connected and simply connected. A structure of rooted tree is defined by the choice of root tail f τ F τ, j(f τ ) = f τ. Its vertex v τ := δ(f τ ) also may be called the root. We define the canonical orientation on the rooted trees: the root tail is oriented away from its vertex (so it is the output); all other flags are oriented towards the root vertex. The remaining tails are called inputs. The output tail of a tree can be grafted to an input tail of another tree. We say that a vertex v is a mother for a vertex v if v lies on an oriented path from v to the root vertex v 0 and the oriented path from v to v consists of a single edge. Given an oriented rooted tree τ, we assign to each vertex v V τ a variety X v P d. To the unique outgoing tail at v we assign a choice of an hyperplane H v X v. To each incoming tail f at v we assign a point p v,f in X v such that p v,f p v,f for f f and with p v,f / H v, for all f at v. We think of an oriented rooted tree τ, with S τ the finite set of incoming tails of τ of cardinality n, as an n ary operation that starts with the varieties X vi P d attached to the input vertices v i, i = 1,..., m n, and glues the hyperplane H vi X vi to the exceptional divisor of the blowup of X wi at the point p wi,f i where w i is the target vertex of the unique outgoing edge of v i and f i is the flag of this edge with (f i ) = w i, ingoing at w i. The operation continues in this way at the next step, by gluing the hyperplanes H wi to the exceptional divisor of the blowups of the projective spaces of the following vertex. At each vertex that has an incoming tail, the corresponding variety acquires a marked point. The variety obtained by this series of operation, when one reaches the root vertex, is the output of τ. It is endowed with n marked points from the incoming tails and with a given hyperplane from the outgoing tail at the root. In the terminology of [6], the output X τ of an oriented rooted tree τ with n incoming tails is a n pointed rooted tree of d dimensional projective spaces. The stability condition for X τ is the requirement that each component of X τ contains at least two distinct markings, which can be either marked points or exceptional divisors. By Proposition of [6], this condition is equivalent to the absence of nontrivial automorphisms. Theorem of [6] defines the variety T d,n as the moduli space of n pointed stable rooted trees of d dimensional projective spaces. In [6] it is shown that the varieties T d,n are naturally embedded in the boundary strata of the Fulton MacPherson compactification of configuration spaces. We will now construct affine torifications on the varieties T d,n using the Fulton MacPherson compactification spaces.

4 4 Yuri I. Manin and Matilde Marcolli Let X be a smooth projective toric variety. A vector bundle E over X is equivariant (or toric) if the torus action on X lifts to an action on E that is linear on the fibers, see [3], [16], [19] for characterizations of these bundles. Lemma 2.1. (Lemma 4.3 of [2]). Let X be a smooth projective toric variety and let E be an equivariant vector bundle on X. Then the projective bundle P(E) admits an affine torification, obtained using the affine torification {U σ } of X as a toric variety, and the compatible affine torifications on the toric varieties P(E Uσ ). This follows from the fact that toric varieties have a natural choice of an affine torification and that, by Proposition 1.13 of [19], the restriction of E to the affine open sets U α is a sum of line bundles. It is shown in [16] that, for a toric bundle that is a sum of line bundles, the projectivization P(E Uα ) is itself a toric variety, hence it has a compatible affine torification. The equivariant gluings on the intersections imply the compatibility of the torifications, which determine an affine torification of P(E), with the projection a torified morphism. Let S be a finite set. We consider the set of rooted trees τ with S τ S and we denote by T d,s the variety constructed as above. We also denote by X[S] the Fulton MacPherson space. We then have the following result (see also Section 4.5 of [2]). Theorem 2.2. The varieties T d,s admit affine torifications induced by the affine torifications of the Fulton MacPherson spaces X[S], with X a toric variety, obtained through the iterated blowup-construction of X[S]. Proof. Let us denote by X[S] the Fulton MacPherson space. We describe its construction in terms of iterated blowups, following the general construction for graph configuration spaces used in [20], [21] and in [5], in the special case of the complete graph. One starts with the product X S of n = #S copies of X and considers all diagonals S X S for all subsets S S, given by S = {x X S x i = x j, i, j S }. Upon identifying the subset S S with the set of vertices of a subgraph Γ S Γ S, where Γ S is the complete graph on the set S of vertices, the diagonal S is identified with a product X V Γ S /Γ S where the quotient graph Γ S /Γ S is obtained by identifying all of Γ S to a single vertex. Consider the set G S of all subgraphs Γ S that are biconnected (that is, that cannot be disconnected by removing the star of any one vertex) and choose an ordering G S = {Γ S 1,..., Γ S N } such that if S i S j then the indices are ordered by i j. By dominant transform of a subvariety under a blowup one means the proper transform if the variety is not contained in the blowup locus and the inverse image otherwise (Definition 2.7 of [20]). It was shown in Theorem 1.3 and Proposition 2.13 of [20] (see also Proposition 2 of [5]) that the sequence of blowups Y (k) with Y (0) = X S and Y (k+1) obtained by blowing up Y (k) along the dominant transform of S k gives Y (N) = X[S], the Fulton MacPherson compactification. Let D(S ) be the divisors on X[S] obtained as iterated dominant transforms of the diagonals S, for Γ S in G S. By Theorem 1.2 of [20] and Proposition 4

5 Moduli Operad over F 1 5 of [5], the intersections D(S k 1 ) D(S k r ) are non-empty if and only if the collection of graphs N = {Γ S k1,..., Γ S kr } forms a G S -nest, that is, it is a set of biconnected subgraphs of type Γ S such that any two subgraphs are either disjoint or they intersect at a single vertex or one is contained as subgraph in the other (see Section 4.3 of [20] and Proposition 3 of [5]). The projectivized normal bundles of the blowup loci have an explicit description in terms of screen configurations, see Section 1 of [10]. At a point x X, if T x = T x (X) denotes the tangent space at x, the screen configuration space for S is given by the projectivization P(T S x /T x ), which can be thought of as the infinitesimal information about the tangent directions to the points that collide on the diagonal, taken up to translation and homothety. As in Sections 1 and 2 of [10] and Theorem 1 of [5], at each step of the iterated blowup construction of X[S], when the dominant transform of a diagonal S is blown up, one can identify the exceptional divisor with the projectivized normal bundle P(N ΓS N,S S S ( S )) for a G S -nest N, or with the projectivized normal bundle P(N X S ( S )). Moreover, these projectivized normal bundles can be described in terms of screen configurations P(N ΓS N,S S S ( S )) P(T S N /T), where the set S N is the set of vertices of the quotient of the graph ΓS obtained by contracting each component of the graph Γ S Γ k S to a vertex, where 1 kr {Γ S,..., Γ k S } = {Γ S N, S S }. 1 kr If X is a toric variety, the tangent bundle T = T (X) and the quotients T S N /T are equivariant (toric) bundles, hence the projectivizations P(T S N /T) admit affine torifications, compatible with the affine torification of the toric variety X S. Thus, the blowup has an affine torification and the projection map to X S is a morphism of affinely torified schemes. The varieties T d,s can be identified with the fibers of the projection π : D(S) X S X S. The map π is a morphism of affinely torified varieties, hence the fiber carries a compatible affine torification. In particular, this gives a construction of affine torifications of T 1,n = M 0,n+1. Remark. The compatibility of the torifications at the various steps in the inductive procedure described above is ensured by the fact that they are all induced by the same torus action on X. At every step one constructs a torification on the screen configuration spaces P(T S /T), which is always induced by the torus action on the torus-equivariant vector bundle T, and all the maps involved are equivariant Counting points over F 1 m. In [6], the Poincaré polynomials of the varieties T d,n are computed, generalizing the result of [26] on the Poincaré polynomial of the moduli spaces M 0,n. This result can be interpreted as giving the counting of

6 6 Yuri I. Manin and Matilde Marcolli points over F 1 m. In fact, the number of points over F 1 can be counted by the limit as q 1 of the function N X (q) that counts points over finite fields F q, eventually normalized by a power of q 1. The value N X (1) for a polynomially countable variety coincides with its Euler characteristic. Similarly, one can make sense of the number of points over extensions F 1 m as the values N X (m+1), see Theorem 4.10 of [8] and Theorem 1 of [9]. Proposition 2.3. For a fixed d, denote by p n,m the number of points of T d,n over F 1 m and form a generating function η m (t) = n 1 p n,m t n. n! This function is a solution of the differential equation (1 + (m + 1) d t (m + 1)κ d (m + 1)η m )η m = 1 + η m. Proof. Theorem and Corollary of [6] shows that, for a fixed d and for #S = n 2, the generating series ψ(q, t) = n 1 P n (q) n! for the Poincaré polynomials P n (q) := P Td,S (q), with P 1 (q) = 1, is the unique solution in t + t 2 Q[q][[t]] to the differential equation where t n, (1 + q 2d t q 2 κ d (q 2 )ψ) t ψ = 1 + ψ, κ d (q 2 ) = q2d 1 q 2 1. Let Z be a smooth projective variety over Z whose class in the Grothendieck group can be written as [Z] = i a il i with the a i non negative integers. For all but finitely many primes p and q = p r, the function that counts points of a finite fields F q is then given by N Z (q) = i a iq i, while for the corresponding complex variety the Poincaré polynomial in a formal variable q is given by P Z (q) = i a iq 2i. Thus, for fixed d, the number of points of T d,n over F 1 m is counted by the values P n (q 1/2 ) q=m+1. The generating function η m (t) of these numbers is a solution of the differential equation in Prop Affine torifications of M 0,n from toric embeddings. We describe here a different possible construction of affine torifications on the moduli spaces M 0,n, using the embeddings of M 0,n in toric varieties as in [13], [14], [17], [29]. Theorem 2.4. The moduli spaces M 0,n have affine torifications induced by embeddings M 0,n X into smooth non complete toric varieties X.

7 Moduli Operad over F 1 7 Proof. In [13] [14], and [29], one considers a simplicial complex with the set of vertices I = {I {1,... n}, 1 I, #I 2, #I c 2} and with simplexes σ if for all I and J in σ either I J or J I or I J = {1,..., n}. The collection of cones associated to the simplexes σ in determine a polyhedral fan in R (n 2) 1, which also arises in tropical geometry as the space of phylogenetic trees [28]. The associated toric variety X is smooth, though not complete. The moduli space M 0,n embeds in X and it intersects the torus T of X in M 0,n. The boundary strata of M 0,n are pullbacks of torus invariant loci in X (see Section 6 of [14] and Section 5 of [13]). The toric variety X admits an affine torification. This induces compatible affine torifications on torus-invariant subvarieties, since the torification is determined by orbits of the torus action. Thus, one obtains an induced affine torification on M 0,n. Question. Does the identity morphism of M 0,n induce an isomorphism of two torifications? 2.4. Blueprint structure on M 0,n. The construction of the toric variety X in [13] [14], and [29] with the embedding M 0,n X, relies on an earlier result of Kapranov realizing M 0,n as a quotient of a Grassmannian. More precisely, in [17], Kapranov showed that the quotient G 0 (2, n)/t of the open cell G 0 (2, n) of points with non-vanishing Plücker coordinates in the Grassmannian G(2, n), by the action of an (n 1)-dimensional torus T is the moduli space M 0,n, and its compactification M 0,n is obtained as the (Chow or Hilbert) quotient of G(2, n) by the action of T. From the point of view of F 1 -geometry, observe that the Plücker embedding of the Grassmannian G(2, n) P (n 2) 1, used to obtain M 0,n in this way, also furnishes G(2, n) with an F 1 structure as blueprint in the sense of [23], [24] (but not as affinely torified varieties), where the blueprint structure (see [24] and Section 5 of [23]) is defined by the congruence R generated by the Plücker relations x ij x kl + x il x jk = x ik x jl for 1 i < j < k < l n. One can use the Plücker coordinates, together with the toric variety construction of [13] [14], and [29], to obtain explicit equations for M 0,n in the Cox ring of the toric variety X, see Theorem 1.2 of [13] and [18]. This can be used to give a blueprint structure on M 0,n. Theorem 2.5. The moduli spaces M 0,n have a blueprint structure O F1 (M 0,n ) = A//R, with the monoid A = F 1 [x I : I I] := { I x n I I } n I 0,

8 8 Yuri I. Manin and Matilde Marcolli where Q[x I, : I I] is the Cox ring of X, and the blueprint relations given by R = S 1 f R A, with R = x I + x I x I : 1 i < j < k < l n, ij I, kl / I il I, jk / I ik I, jl / I and S 1 f R the localization with respect to the submonoid generated by the element f = I x I. Proof. Proposition 2.1 of [13] gives a general method for producing explicit equations for quotients of subvarieties of a torus by the action of a subtorus. Theorem 3.2 of [13] uses this result to obtain explicit equations for Chow and Hilbert quotients of T d -equivariant subschemes of projective spaces P m. Theorem 6.3 of [13] then obtains explicit equations for M 0,n inside the toric variety X starting with the Plücker relations on the Grassmannian G(2, n) and the quotient description of M 0,n obtained in [17]. More precisely, the equations for M 0,n are obtained by homogenizing the Plücker relations with respect to the grading in the Cox ring of X and then saturating by the product of the variables in the Cox ring. With the notation I : J for the saturation of an ideal I by J, the equations for M 0,n are given by (Theorem 6.3 of [13]) x I x I + x I : ij I, kl / I ik I, jl / I il I, jk / I ( ) x I, I where Q[x I, : I I] is the Cox ring of X, and where i, j, k, l satisfy 1 i < j < k < l n. In general, let A be a polynomial ring and I an ideal and let J = (f) be the ideal generated by an element f. Then the saturation I : J = I f A, where I f is the localization of I at f. Thus, we can write the ideal of M 0,n in terms of localizations. As in Section 4 of [23], we can consider the blueprint B = A//R, with the monoid A = F 1 [x I : I I] and the blueprint relations R given in the statement. As shown in Section 1.13 of [24], blueprints admit localizations with respect to submonoids of A. Thus, given the element f = I x I, and S f the submonoid of A generated by f, we can consider the localization S 1 f R = ij I, kl / I x I f a + I il I, jk / I x I f a I x I f a I ik I, jl / I : 1 i < j < k < l n, where the localized blueprint relation S 1 f R lives in the localization S 1 f A A S f, given by the set of equivalence classes (denoted a/f k ) of elements (a, f k ) with the relation (a, f k ) (b, f l ) when f k+m b = f l+m a for some m. The blueprint relations R = S 1 f R A then give the blueprint structure of M 0,n.

9 Moduli Operad over F Remarks on higher genera. The moduli spaces M g,n of stable curves of higher genus with marked points have Deligne Mumford compactifications M g,n, with natural morphisms between them, similar to the genus zero case: inclusions of boundary strata M g1,n 1+1 M g2,n 2+1 M g1+g 2,n 1+n 2 and forgetting markings (and stabilizing) M g,n M g,n 1, as well as morphisms arising from gluing two marked points together, M g,n+2 M g+1,n. However, M g,n are generally only stacks rather than schemes. One does not expect the higher genus moduli spaces to carry F 1 structures. However, one can consider interesting sub loci of these moduli spaces, M 0 g,n, parametrizing curves whose irreducible components are all rational. These stacks can be made components of an operad, and at least some covers of them admit a compatible F 1 structure. In order to complete this picture, the basics of DM stacks theory over F 1 must be developed first Blueprints and the M 0 g,n strata. The locus M 0 g,n of rational curves in the higher genus moduli space M g,n can be described, as explained in [12], as the image of a finite map R : M 0,2g+n M g,n, obtained in the following way. The locus M 0 g,n is the closure of the locus of irreducible g-nodal curves. These curves have normalization given by a smooth rational curve with 2g + n marked points. One can then consider the subgroup G S 2g of permutations of these 2g additional marked points that commute with the product (12)(34) (2g 1 2g) of g transpositions, so that the normalization of M 0 g,n can be identified with the quotient M 0,2g+n /G. Recall that a blueprint A//R is constructed by considering a commutative multiplicative monoid A and the associated semiring N[A], together with a set of relations R N[A] N[A], written as relations a i b j, for ( a i, b j ) R. We say that a group G acts on a blueprint A//R by automorphisms if it acts by automorphisms of the monoid A and the induced diagonal action on N[A] N[A] preserves the set of blueprint relations R. Lemma 2.6. The action of G on M 0,2g+n induces an action by automorphisms of the blueprint O F1 (M 0,2g+n ).

10 10 Yuri I. Manin and Matilde Marcolli Proof. In general, the action of the symmetric group S n on M 0,n by permutation of the marked points induces an action by automorphism on the commutative monoid A = F 1 [x I : I I] described above, by correspondingly permuting the coordinates x I. This action fixes the element f = I x I and perserves the set of blueprint relations R, because it corresponds to the action on the set of Plücker relations by permuting matrix columns. Thus, the subgroup G S 2g S 2g+n also acts by automorphisms of the monoid A of M 0,2g+n preserving the blueprint relations, hence as automorphisms of O F1 (M 0,2g+n ). In order to obtain F 1 -data for the quotient M 0,2g+n /G, we suggest an approach that uses the point of view of noncommutative geometry, replacing the quotient operation by a crossed product by the group of symmetries, at the level of the associated algebraic structure. This point of view suggests introducing a notion of (non-commutative) crossed product blueprints. Definition 2.7. Let A//R be a blueprint with A a commutative multiplicative monoid and R a set of blueprint relations and let G a group of automorphisms of A//R. The monoid crossed product A G is the multiplicative (non-commutative) monoid with elements of the form (a, g) with a A and g G with product (a, g)(a, g ) = (ag(a ), gg ). The semiring crossed product N[A] G is given by all finite formal sums (a i, g i ) with a i A and g i G with multiplication (a i, g i )(a j, g j ) = (a i g i (a j ), g i g j ). Let R G (N[A] G) (N[A] G) be the set of elements (( a i, g), ( b j, g)), with ( a i, b j ) R and g G. The crossed product (A//R) G is defined as the pair (A G, R G ). Lemma 2.8. The action of the symmetric group S n on the moduli space M 0,n determines a crossed product blueprint O F1 (M 0,n ) GL n (F 1 ). Proof. This is an immediate consequence of Lemma 2.6, Definition 2.7 and the identification S n = GL n (F 1 ). We can then use this notion of crossed product blueprint to associate F 1 -data to the strata M 0 g,n of the higher genus moduli spaces M g,n. Proposition 2.9. The normalization of M 0 g,n has an associated crossed product blueprint structure O F1 (M 0,2g+n ) G, with G S 2g the subgroup of permutations that commute with the product of transpositions (12)(34) (2g 1 2g). Proof. Again, this is an immediate consequence of Lemma 2.6 and Definition 2.7. As in noncommutative geometry, the use of crossed product structures is a convenient replacement for the quotient M 0,2g+n /G.

11 Moduli Operad over F Structure morphisms of operads We show that the constructions of torifications described in the previous section are compatible with the operad structures. Let an operad P, in the symmetric monoidal category of varieties over Z with Cartesian product be given. Its descent data to F 1 consist of affine torifications such that the composition operations P(n) P(m 1 ) P(m n ) P(m m n ) and the structure actions of symmetric groups are morphisms of affinely torified varieties. This is a tourist class description of [26]. A more systematic treatment requires the explicit introduction of a category of labeled graphs as in [4] Categories of trees and operads. We consider a category Γ, whose objects are finite disjoint unions of oriented rooted trees. The morphisms are generated by edge contractions and graftings. The grafting of an oriented tree τ to another oriented tree σ is realized by the morphism h : τ σ τ# v0,wσ, where the involution j h matches the outgoing tail of the root vertex v 0 of τ with an ingoing tail of a vertex w of σ. The edge contractions are given by morphisms h e : τ τ/e, where the edge e is a j τ -orbit e = {f, f } of flags f, f F τ and F τ \ h F e (F τ/e ) = {f, f }, h 1 e,f : hf e (F τ/e ) F τ/e the identity and h e,v : V τ V τ/e mapping τ (f) and τ (f ) to the same vertex in τ/e. It is shown in Section IV.2 of [27] that the datum of an operad is equivalent to a monoidal functor M from a category of trees (forests) with the symmetric monoidal structure given by disjoint union and morphisms generated by graftings and edge contractions, to a symmetric monoidal category (C, ), with the condition that M(τ) = v Vτ M(τ v ), where τ v is the star of the vertex v, see Proposition IV of [27]. The operad composition is identified with the image M(ψ) of the morphism ψ that assigns to a disjoint union of corollas τ τ 1 τ n the corolla obtained by first grafting the outgoing tails of the component τ k to the k th ingoing tail of τ and then contracting all the edges Operad structure on T d,n and torifications. The varieties T d,n have natural morphisms defining an operad structure that generalizes the operad of M 0,n. Theorem 3.1. For each fixed d 1, there are morphisms of the following form, which determine an an operad T d : (1) isomorphisms: T d,s Td,S for S S, functorial wrt the bijections of labelling sets. (2) embeddings: T d,s T d,s\s { } T d,s, for S S with #S 2.

12 12 Yuri I. Manin and Matilde Marcolli (3) forgetful morphisms: T d,s T d,s for S S with #S 2. These morphisms satisfy the standard identities. Proof. The existence of morphisms of the form (1) is clear by construction. The cases (2) and (3) follow from the boundary stratification of these varieties constructed in Theorem of [6]. In fact, the boundary of a variety T d,s is given by smooth normal crossings divisors: given any proper subset S S, there is a nonsingular divisor T d,s (S ) T d,s. These divisors meet transversely and the only non empty intersections T d,s (S 1 ) T d,s (S r ) happen when the sets S k are nested (each pair is either disjoint or one is a subset of the other). The divisors satisfy T d,s (S ) T d,s T d,s\s { }. This gives the morphisms (2) coming from the inclusion of the strata. In terms of morphisms of oriented rooted trees, these correspond to the morphisms that graft the outgoing tail of the first tree with set of incoming tails identified with S to the incoming tail marked by in the second tree. The forgetful morphisms (3) come from the construction of T d,s { } from T d,s via a sequence of iterated blowups, as in Theorem of [6]. The composition of the projections of this sequence of blowups give the forgetful morphism T d,s { } T d,s. In terms of rooted trees of projective spaces, these correspond to forgetting some of the marked points and contracting the resulting unstable components. Using the functorial characterization of operads given in Proposition IV of [27], let (C, ) be the symmetric monoidal category of algebraic varieties with the Cartesian product, and let (Γ, ) be the category of oriented rooted forests with disjoint union. The embeddings of the strata determine the morphisms T d (ψ), where ψ is the morphism of oriented rooted trees that assigns to a disjoint union of oriented corollas τ τ 1 τ n, where each corolla has only one outgoing tail, the corolla obtained by first grafting the outgoing tails of the component τ k to the k th ingoing tail of τ and then contracting all the edges. This assignment determines the operad composition operations T d,s T d,s1 T d,sn T d,s1 S n, where n = #S is the number of incoming tails of the trees of projective spaces parameterized by T d,s. Theorem 3.2. For each fixed d 1, the morphisms (1), (2), (3) of Theorem 3.1 are torified, with respect to the torification constructed in Theorem 2.2. Proof. The tori in the torification of T d,s are indexed by an indexing of the G S nests and, for each G S -nest N, by an indexing of the tori in the torification of P(T S N /T). The isomorphisms (1) are then torified, by taking as map of the indexing sets φ : J J the isomorphism G S G S induced by the isomorphism S S and for matching N and N under this map, a bijection of the indexing sets of the tori in P(T S N /T) P(T S N /T). The boundary of the Fulton MacPherson compactification is a union of divisors D(S ) given by the dominant transforms in the iterated blowup of the diagonals S of biconnected graphs Γ S. The affine

13 Moduli Operad over F 1 13 torification on X[S] is by construction compatible with restriction to the strata, since it is constructed using torifications of the spaces of screen configurations P(N ΓS N,S S ( S S )), so that the boundary strata D(S ) are unions of tori of the torification and the inclusion of strata preserves the torification. Thus, morphisms (2) arising from inclusions of strata are morphisms of affinely torified varieties. Similarly, morphisms (3) are given by the projection maps of the iterated blowups, which also, by construction, are morphisms of affinely torified varieties. Corollary 3.3. The operads T d are endowed with descent data to F 1. Proof. This follows directly from the previous theorem, since the operad composition operators T d,s T d,s1 T d,sn T d,s1 S n, with n = #S, are obtained from the morphisms of type (2) of Theorem 3.2, which are compatible with the torification Toric embeddings and operadic structure on {M 0,n }. We now consider the other construction of torifications on M 0,n, described in Theorem 2.4, and we show that it is also compatible with the operad structure. Theorem 3.4. Let M(n) = M 0,n+1. The composition morphisms of the operad M(n) M(m 1 ) M(m n ) M(m m n ) are morphisms of affinely torified varieties, with respect to the torifications of Theorem 2.4. The same is true about symmetric group actions. Proof. Consider an affine torification of the toric variety X of [13] [14], [29]. The embedding M 0,n X induces a compatible affine torifications, which is compatible with the inclusion of the boundary strata M 0,n1+1 M 0,n2+1 M 0,n, for n 1 + n 2 = n, since the boundary strata of M 0,n are pullbacks of torus-invariant loci in X (Proposition 6.2 of [14]). Thus, the composition operations that define the operad structure, which are inclusions of boundary strata, are compatible with the torification. References [1] K. Behrend, Yu.I. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J. Vol.85 (1996) N.1, [2] D. Bejleri, M. Marcolli, Quantum field theory over F 1, arxiv:

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15 Moduli Operad over F 1 15 [25] Yu. I. Manin, Cyclotomy and analytic geometry over F 1, in Quanta of maths, , Clay Math. Proc., 11, Amer. Math. Soc., [26] Yu. I. Manin, Generating functions in algebraic geometry and sums over trees, in The moduli space of curves (Texel Island), Progr. Math. Vol. 129, , Birkhäuser, [27] Yu. I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, Colloquium Publications, Vol.47, American Mathematical Society, [28] D. Speyer, B. Sturmfels, The tropical Grassmannian, Adv. Geom. Vol.4 (2004) N.3, [29] J. Tevelev, Compactifications of subvarieties of tori, American J. Math. Vol.129 (2007) Yuri I. Manin, Max Planck Institut für Mathematik, Vivatsgasse 7, Bonn, D-53111, Germany manin@mpim-bonn.mpg.de Matilde Marcolli, Department of Mathematics, California Institute of Technology, 1200 E California Boulevard, Pasadena, CA 91125, USA matilde@caltech.edu