Compactifications of M 0,n associated with Alexander self-dual complexes: Chow ring, ψ -classes and intersection numbers
aa r X i v : . [ m a t h . G T ] A ug COMPACTIFICATIONS OF M ,n ASSOCIATED WITH ALEXANDER SELF DUALCOMPLEXES: CHOW RING, ψ -CLASSES, AND INTERSECTION NUMBERS ILIA NEKRASOV AND GAIANE PANINA Abstract. An Alexander self-dual complex gives rise to a compactification of M ,n , called ASD com-pactification , which is a smooth algebraic variety. ASD compactifications include (but are not exhaustedby) the polygon spaces , or the moduli spaces of flexible polygons. We present an explicit description ofthe Chow rings of ASD compactifications. We study the analogs of Kontsevich’s tautological bundles,compute their Chern classes, compute top intersections of the Chern classes, and derive a recursion for theintersection numbers.
Alexander self dual complex, modular compactification, tautological ring, Chern class, Chow ring1.
Introduction
The moduli space of n -punctured rational curves M ,n and its various compactifications is a clas-sical object, bringing together algebraic geometry, combinatorics, and topological robotics. Recently,D.I.Smyth [1] classified all modular compactifications of M ,n . We make use of an interplay betweendifferent compactifications, and: • describe the classification in terms of (what we call) preASD simplicial complexes ; • describe the Chow rings of the compactifications arising from Alexander self-dual complexes (ASDcompactifications); • compute for ASD compactifications the associated Kontsevich’s ψ -classes , their top monomials,and give a recurrence relation for the top monomials.Oversimplifying, the main approach is as follows. Some (but not all) compactifications are the well-studied polygon spaces , that is, moduli spaces of flexible polygons. A polygon space corresponds to a threshold Alexander self-dual complex. Its cohomology ring (which equals the Chow ring) is known dueto J.-C. Hausmann and A. Knutson [2], and A. Klyachko[10]. The paper [3] gives a computation-friendlypresentation of the ring. Due to Smyth [1], all the modular compactifications correspond to preASDcomplexes , that is, to those complexes that are contained in an ASD complex. A removal of a facet of apreASD complex amounts to a blow up of the associated compactification. Each ASD compactification isachievable from a threshold ASD compactification by a sequence of blow ups and blow downs. Since thechanges in the Chow ring are controllable, one can start with a polygon space, and then (by elementarysteps) reach any of the ASD compactifications and describe its Chow ring (Theorem 26).M. Kontsevich’s ψ -classes [4] arise here in a standard way. Their computation of is a mere modificationof the Chern number count for the tangent bundle over S (a classical exercise in a topology course). Therecursion (Theorem 36) and the top monomial counts (Theorem 37) follow.It is worthy mentioning that a disguised compactification by simple games, i.e., ASD complexes, isdiscussed from a combinatorial viewpoint in [5].Now let us give a very brief overview of moduli compactifications of M ,n . A compactification by asmooth variety is very desirable since it makes intersection theory applicable. We also expect that (1) acompactification is modular , that is, itself is the moduli space of some curves and marked points lying onit, and (2) the complement of M ,n (the “boundary”) is a divisor.The space M ,n is viewed as the configuration space of n distinct marked points (“particles”) living inthe complex projective plane. The space M ,n is non-compact due to forbidden collisions of the marked Chebyshev laboratory, Department of Math. and Mech., St. Petersburg State University PDMI RAS, St. Petersburg State University
E-mail addresses : [email protected], [email protected] . M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES points. Therefore, each compactification should suggest an answer to the question: what happens whentwo (or more) marked points tend to each other? There exist two possible choices: either one allowssome (not too many!) points to coincide, either one applies a blow up. It is important that the blow upsamount to adding points that correspond to n -punctured nodal curves of arithmetic genus zero.A compactification obtained by blow ups only is the celebrated Deligne–Mumford compactification .If one avoids blow ups and allows (some carefully chosen) collections of points to coincide, one gets anASD-compactification; among them are the polygon spaces . Diverse combinations of these two options(in certain cases one allows points to collide, in other cases one applies a blow up) are also possible; thecomplete classification is due to [1].Now let us be more precise and look at the compactifications in more detail.1.1.
Deligne–Mumford compactification.Definition 1. [6] Let B be a scheme. A family of rational nodal curves with n marked points over B is • a flat proper morphism π : C → B whose geometric fibers E • are nodal connected curves ofarithmetic genus zero, and • a set of sections ( s , . . . , s n ) that do not intersect nodal points of geometric fibers.In this language, the sections correspond to marked points. The above condition means that anodal point of a curve may not be marked.A family ( π : C → B ; s , . . . , s n ) is stable if the divisor K π + s + · · · + s n is π -relatively ample.Let us rephrase this condition: a family ( π : C → B ; s , . . . , s n ) is stable if each irreducible componentof each geometric fiber has at least three special points (nodal points and points of the sections s i ). Theorem 2. [6] (1) There exists a smooth and proper over Z stack M ,n , representing the moduli functorof stable rational curves. Corresponding moduli scheme is a smooth projective variety over Z .(2) The compactification equals the moduli space for n -punctured stable curves of arithmetic genus zerowith n marked points. A stable curve is a curve of arithmetic genus zero with at worst nodal singularitiesand finite automorphism group. This means that (i) every irreducible component has at least three markedor nodal points, and (ii) no marked point is a nodal point. The Deligne-Mumford compactification has a natural stratification by stable trees with n leaves. A stable tree with n leaves is a tree with exactly n leaves enumerated by elements of [ n ] = { , ..., n } suchthat each vertice is at least trivalent.Here and in the sequel, we use the following notation : by vertices of a tree we mean all the vertices(in the usual graph-theoretical sense) excluding the leaves. A bold edge is an edge connecting two vertices(see Figure 1).The initial space M ,n is a stratum corresponding to the one-vertex tree. Two-vertex trees (Fig.1(b))are in a bijection with bipartitions of the set [ n ]: T ⊔ T c = [ n ] s.t. | T | , | T c | >
1. We denote the closure ofthe corresponding stratum by D T . The latter are important since the (Poincar´e duals of) closures of thestrata generate the Chow ring A ∗ ( M ,n ): Theorem 3. [7, Theorem 1] The Chow ring A ∗ ( M ,n ) is isomorphic to the polynomial ring Z [ D T : T ⊂ [ n ]; | T | > , | T c | > factorized by the relations: (1) D T = D T c ; (2) D T D S = 0 unless S ⊂ T or T ⊂ S or S ⊂ T c or T ⊂ S c ; (3) For any distinct elements i, j, k, l ∈ [ n ] : X i,j ∈ T ; k,l T D T = X i,k ∈ T ; j,l T D T = X i,l ∈ T ; j,k T D T OMPACTIFICATIONS OF M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES 3 (a)(b)
Figure 1.
Stable nodal curves (left) and the corresponding trees (right)1.2.
Weighted compactifications.
The next breakthrough step was done by B. Hassett in [8].Define a weight data as an element A = ( a , . . . , a n ) ∈ R n such that • < a i ≤ i ∈ [ n ], • a + · · · + a n > Definition 4.
Let B be a scheme. A family of nodal curves with n marked points ( π : C → B ; s , . . . , s n )is A –stable if(1) K π + a s + · · · + a n s n is π -relatively ample,(2) whenever the sections { s i } i ∈ I intersect for some I ⊂ [ n ], one has P i ∈ I a i < Theorem 5. [8, Theorem 2.1] For any weight data A there exist a connected Deligne–Mumford stack M , A smooth and proper over Z , representing the moduli functor of A –stable rational curves. The corre-sponding moduli scheme is a smooth projective variety over Z . The Deligne–Mumford compactification arises as a special case for the weight data (1 , . . . , M , A depend on A ? Pursuing thisquestion, let us consider the space of parameters: A n = ( A ∈ R n : 0 < a i ≤ , X i a i > ) ⊂ R n . The hyperplanes P i ∈ I a i = 1, I ⊂ [ n ] , | I | ≥
2, (called walls ) cut the polytope A n into chambers . TheHassett compactification depends on a chamber only [8, Proposition 5.1].Combinatorial stratification of the space M , A looks similarly to that of the Deligne–Mumford’s withthe only difference — some of the marked points now can coincide [9].More precisely, a weighted tree ( γ, I ) is an ordered k -partition I ⊔ · · · ⊔ I k = [ n ] and a tree γ with k ordered leaves marked by elements of the partition such that (1) P j ∈ I m a j ≤ m , and (2) foreach vertex, the number of emanating bold edges plus the total weight is greater than 2. Open strataare enumerated by weighted trees: the stratum of the space M , A corresponding to a weighted tree( γ, I ) consists of curves whose irreducible components form the tree γ and collisions of sections form thepartition I . Closure of this stratum is denoted by D ( γ,I ) . COMPACTIFICATIONS OF M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES
Polygon spaces as compactifications of M ,n . Assume that an n -tuple of positive real numbers L = ( l , ..., l n ) is fixed. We associate with it a flexible polygon , that is, n rigid bars of lengths l i connectedin a cyclic chain by revolving joints. A configuration of L is an n -tuple of points ( q , ..., q n ) , q i ∈ R , with | q i q i +1 | = l i , | q n q | = l n .The following two definitions for the polygon space , or the moduli space of the flexible polygon areequivalent: Definition 6. [2]I. The moduli space M L is a set of all configurations of L modulo orientation preserving isometriesof R .II. Alternatively, the space M L equals the quotient of the space ( ( u , ..., u n ) ∈ ( S ) n : n X i =1 l i u i = 0 ) by the diagonal action of the group SO ( R ).The second definition shows that the space M L does not depend on the ordering of { l , ..., l n } ; however,it does depend on the values of l i .Let us consider the parameter space ( l , . . . , l n ) ∈ R n> : l i < X j = i l j for i = 1 , . . . , n . This space is cut into open chambers by walls . The latter are hyperplanes with defining equations X i ∈ I l i = X j / ∈ I l j . The diffeomorphic type of M L depends only on the chamber containing L . For a point L lying strictlyinside some chamber, the space M L is a smooth (2 n − generic . Definition 7.
For a generic length vector L , we call a subset J ⊂ [ n ] long if X i ∈ J l i > X i/ ∈ J l i . Otherwise, J is called short . The set of all short sets we denote by SHORT ( L ).Each subset of a short set is also short, therefore SHORT ( L ) is a (threshold Alexander self-dual)simplicial complex. Rephrasing the above, the diffeomorphic type of M L is defined by the simplicialcomplex SHORT ( L ). 2. ASD and preASD compactifications
ASD and preASD simplicial complexes.
Simplicial complexes provide a necessary combinatorialframework for the description of the category of smooth modular compactifications of M ,n .A simplicial complex (a complex, for short) K is a subset of 2 [ n ] with the hereditary property: A ⊂ B ∈ K implies A ∈ K . Elements of K are called faces of the complex. Elements of 2 [ n ] \ K are called non-faces . The maximal (by inclusion) faces are called facets .We assume that the set of 0-faces (the set of vertices) of a complex is [ n ]. The complex 2 [ n ] is denotedby ∆ n − . Its k -skeleton is denoted by ∆ kn − . In particular, ∆ n − n − is the boundary complex of the simplex∆ n − . Definition 8.
For a complex K ⊂ [ n ] , its Alexander dual is the simplicial complex K ◦ := { A ⊂ [ n ] : A c K } = { A c : A ∈ [ n ] \ K } . OMPACTIFICATIONS OF M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES 5
Here and in the sequel, A c = [ n ] \ A is the complement of A . A complex K is Alexander self-dual (anASD complex) if K ◦ = K . A pre Alexander self-dual (a pre ASD) complex is a complex contained insome ASD complex.In other words, ASD complexes (pre ASD complexes, respectively) are characterized by the condition:for any partition [ n ] = A ⊔ B , exactly one (at most one, respectively) of A , B is a face.Some ASD complexes are threshold complexes : they equal SHORT ( L ) for some generic weight vectors L (Section 1.3). It is known that threshold ASD complexes exhaust all ASD complexes for n ≤ n this is no longer true. Moreover, for n → ∞ the percentage of threshold ASDcomplexes tends to zero.To produce new examples of ASD complexes, we use flips : Definition 9. [5] For an ASD complex K and a facet A ∈ K we build a new ASD complexflip A ( K ) := ( K \ A ) ∪ A c . It is easy to see that
Proposition 10. (1) [5] Inverse of a flip is also some flip. (2) [5] Any two ASD complexes are connectedby a sequence of flips. (3) For any ASD complex K there exists a threshold ASD complex K ′ that can beobtained from K by a sequence of flips with some A i ⊂ [ n ] such that | A i | > , | A ci | > .Proof. We prove (3). It is sufficient to show that for any ASD complex, there exists a threshold ASDcomplex with the same collection of 2-element non-faces. For this, let us observe that any two non-facesof an ASD complex necessarily intersect. Therefore, all possible collections of 2-element non-faces of anASD complex (up to renumbering) are:(1) empty set;(2) (12) , (23) , (31);(3) (12) , (13) , . . . , (1 k ).It is easy to find appropriate threshold ASD complexes for all these cases. (cid:3) ASD complexes appear in the game theory literature as “simple games with constant sum” (see [11]).One imagines n players and all possible ways of partitioning them into two teams. The teams compete,and a team looses if it belongs to K . In the language of flexible polygons, a short set is a loosing team. Contraction, or freezing operation.
Given an ASD complex K , let us build a new ASD complex K ( ij ) with n − { , ..., b i, ..., b j, ..., n, ( i, j ) } by contracting the edge { i, j } ∈ K , or freezing i and j together.The formal definition is: for A ⊂ { , ..., b i, ..., b j, ..., n } , A ∈ K ( ij ) iff A ∈ K , and A ∪ { ( ij ) } ∈ K ( ij ) iff A ∪ { i, j } ∈ K .Contraction K I of any other face I ∈ K is defined analogously.Informally, in the language of simple game, contraction of an edge means making one player out of two.In the language of flexible polygons, “freezing” means producing one new edge out of two old ones (thelengths sum up).2.2. Smooth extremal assignment compactifications.
Now we review the results of [1] and [12],and indicate a relation with preASD complexes.For a scheme B , consider the space U ,n ( B ) of all flat, proper, finitely-presented morphisms π : C → B with n sections { s i } i ∈ [ n ] , and connected, reduced, one-dimensional geometric fibers of genus zero. Denoteby V ,n the irreducible component of U ,n that contains M ,n . Definition 11.
A modular compactification of M ,n is an open substack X ⊂ V ,n that is proper over Z . A modular compactification is stable if every geometric point ( π : C → B ; s , . . . , s n ) is stable. Wecall a modular compactification smooth if it is a smooth algebraic variety. COMPACTIFICATIONS OF M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES
12 3 4 1 2 3 45 5
Figure 2.
Contraction of { , } in a simplicial complex Definition 12.
A smooth extremal assignment Z over M ,n is an assignment to each stable tree with n leaves a subset of its vertices γ
7→ Z ( γ ) ⊂ V ert ( γ )such that:(1) for any tree γ , the assignment is a proper subset of vertices: Z ( γ ) $ V ert ( γ ),(2) for any contraction γ τ with { v i } i ∈ I ⊂ V ert ( γ ) contracted to v ∈ V ert ( τ ), we have v i ∈ Z ( γ )for all i ∈ I if and only if v ∈ Z ( τ ).(3) for any tree γ and v ∈ Z ( γ ) there exists a two-vertex tree γ ′ and v ′ ∈ Z ( γ ′ ) such that γ ′ γ and v ′ v. Definition 13.
Assume that Z is a smooth extremal assignment. A curve ( π : C → B ; s , . . . , s n ) is Z –stable if it can be obtained from some Deligne–Mumford stable curve ( π ′ : C ′ → B ′ ; s ′ , . . . , s ′ n ) by(maximal) blowing down irreducible components of the curve C ′ corresponding to the vertices from theset Z ( γ ( C ′ )).A smooth assignment is completely defined by its value on two-vertex stable trees with n leaves. Thelatter bijectively correspond to unordered partitions A ⊔ A c = [ n ] with | A | , | A c | >
1: sets A and A c areaffixed to two vertices of the tree. The first condition of Definition 13 implies that no more than one of A and A c is “assigned”. One concludes that preASD complexes are in bijection with smooth assignments.All possible modular compactifications of M ,n are parametrized by smooth extremal assignments: Theorem 14. [1, Theorems 1.9 & 1.21] and [12, Theorem 1.3] • For any smooth extremal assignment Z of M ,n , or equivalently, for any preASD complex K ,there exists a stack M , Z = M ,K ⊂ V ,n parameterizing all Z –stable rational curves. • For any smooth modular compactification
X ⊂ V ,n , there exist a smooth extremal assignment Z (a preASD complex K ) such that X = M , Z = M ,K . There are two different ways to look at a moduli spaces. In the present paper we look at the modulispace as at a smooth algebraic variety equipped with n sections ( fine moduli space ). The other way is tolook at it as at a smooth algebraic variety ( coarse moduli space ). Different preASD complexes give rise todifferent fine moduli spaces. However, two different complexes can yield isomorphic coarse moduli spaces.Indeed, consider two preASD complexes K and K ∪ { ij } (we abbreviate the latter as K + ( ij )),assuming that { ij } / ∈ K . The corresponding algebraic varieties M ,K and M ,K +( ij ) are isomorphic. A OMPACTIFICATIONS OF M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES 7 vivid explanation is: to let a couple of marked points to collide is the same as to add a nodal curve withthese two points sitting alone on an irreducible component. Indeed, this irreducible component wouldhave exactly three special points, and PSL acts transitively on triples. Theorem 15. [12, Statements 7.6–7.10] The set of smooth modular compactifications of M ,n is in abijection with objects of the preASD n / ∼ , where K ∼ L whenever K \ L and L \ K consist of two-elementsets only. Example 16.
PreASD complexes and corresponding compactifications.(1) the 0-skeleton ∆ n − = [ n ] of the simplex ∆ n − corresponds to the Deligne–Mumford compactifi-cation;(2) the complex P n := pt ⊔ ∆ n − n − (disjoint union of a point and the boundary of a simplex ∆ n − )is ASD. It corresponds to the Hassett weights (1 , ε, . . . , ε ); this compactification is isomorphic to P n − ;(3) the Losev–Manin compactification M LM ,n [13] corresponds to the weights (1 , , ε, . . . , ε ) and to thecomplex pt ⊔ pt ⊔ ∆ n − ;(4) the space ( P ) n − corresponds to weights (1 , , , ε, . . . , ε ), and to the complex pt ⊔ pt ⊔ pt ⊔ ∆ n − .2.3. ASD compactifications via stable point configurations.
ASD compactifications can be ex-plained in a self-contained way, without referring to [1].Fix an ASD complex K and consider configurations of n (not necessarily all distinct) points p , ..., p n inthe projective line. A configuration is called stable if the index set of each collection of coinciding pointsbelongs to K . That is, whenever p i = ... = p i k , we have { i , ..., i k } ∈ K .Denote by ST ABLE ( K ) the space of stable configurations in the complex projective line. The groupPSL ( C ) acts naturally on this space. Set M ,K := ST ABLE ( K ) / PSL ( C ) . If K is a threshold complex, that is, arises from some flexible polygon L , then the space M ,K isisomorphic to the polygon space M L [10].Although the next theorem fits in a broader context of [1], we give here its elementary proof. Theorem 17.
The space M ,K is a compact smooth variety with a natural complex structure.Proof. Smoothness.
For a distinct triple of indices i, j, k ∈ [ n ], denote by U i,j,k the subset of M ,K defined by p i = p j , p j = p k , and p i = p k . For each of U i,j,k , we get rid of the action of the group PSL ( C ),setting U i,j,k = (cid:8) ( p , ..., p n ) ∈ M ,K : p i = 0 , p j = 1 , and p k = ∞ (cid:9) . Clearly, each of the charts U i,j,k is an open smooth manifold. Since all the U i,j,k cover M ,K , smoothnessis proven. Compactness.
Let us show that each sequence of n -tuples has a converging subsequence.Assume the contrary. Without loss of generality, we may assume that the sequence ( p i = 0 , p i =1 , p i = ∞ , p i , ..., p in ) ∞ i =1 has no converging subsequence. We may assume that for some set I / ∈ K , all p ij with j ∈ I converge to a common point. We say that we have a collapsing long set I . This notion dependson the choice of a chart. We may assume that our collapsing long set has the minimal cardinality amongall long sets that can collapse without a limit (that is, violate compactness) for this complex K . We mayassume that I = { , , , ..., k } .This long set can contain at most one of the points p , p , p . We consider the case when it contains p ; other cases are treated similarly.That is, all the points p i , ..., p ik tend to ∞ . Denote by C i the circle with the minimal radius embracingthe points p i = ∞ , p i , p i , ..., p ik . The circle contains at least two points of p i , ..., p ik , p = ∞ . Applya transform φ i ∈ PSL ( C ) which turns the radius of C i to 1, and keeps at least two of the points COMPACTIFICATIONS OF M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES p i , ..., p ik , p = ∞ away from each other. In this new chart the cardinality of the collapsing long set getssmaller. A contradiction to the minimality assumption. (cid:3) A natural question is: what if one takes a simplicial complex (not a self-dual one), and cooks theanalogous quotient space. Some heuristics are: if the complex contains simultaneously some set A and itscomplement [ n ] \ A , we have a stable tuple with a non-trivial stabilizer in PSL ( C ), so the factor has anatural nontrivial orbifold structure. If a simplicial complex is smaller than some ASD complex K ′ , andtherefore, we get a proper open subset of M ,K ′ , that is, we lose compactness.2.4. Perfect cycles.
Assume that we have an ASD complex K and the associated compactification M ,K . Let K I be the contraction of some face I ∈ K . Since the variety M ,K I naturally embeds in M ,K , the contraction procedure gives rise to a number of subvarieties of M ,K . These varieties (1) “lieon the boundary” and (2) generate the Chow ring (Theorem 26 ). Let us look at them in more detail.An elementary perfect cycle ( ij ) = ( ij ) K ⊂ M ,K is defined as( ij ) = ( ij ) K = { ( p , ..., p n ) ∈ M ,K : p i = p j } . Let [ n ] = A ⊔ ... ⊔ A k be an unordered partition of [ n ]. A perfect cycle associated to the partition( A ) · ... · ( A k ) = ( A ) K · ... · ( A k ) K == { ( p , ..., p n ) ∈ M ,K : i, j ∈ A m ⇒ p i = p j } . Each perfect cycle is isomorphic to M ,K ′ for some complex K ′ obtained from K by a series of con-tractions.Singletons play no role, so we omit all one-element sets A i from our notation. Consequently, all theperfect cycles are labeled by partitions of some subset of [ n ] such that all the A i have at least two elements.Note that for arbitrary A ∈ K , the complex K A might be ill-defined. This happens if A / ∈ K . In thiscase the associated perfect cycle ( A ) is empty.For each perfect cycle there is an associated Poincar´e dual element in the cohomology ring. These dualelements we denote by the same symbols as the perfect cycles.The following rules allow to compute the cup-product of perfect cycles: Proposition 18. (1)
Let A and B be disjoint subsets of [ n ] . Then (a) ( A ) ⌣ ( B ) = ( A ) · ( B ) . (b) ( Ai ) ⌣ ( Bi ) = ( ABi ) . (2) For
A / ∈ K , we have ( A ) = 0 . If one of A k is a non-face of K , then ( A ) · ... · ( A k ) = 0 . (3) The four-term relation: ( ij ) + ( kl ) = ( jk ) + ( il ) holds for any distinct i, j, k, l .Proof. In the cases (1) and (2) we have a transversal intersection of holomorphically embedded complexvarieties. The item (3) will be proven in Theorem 26. (cid:3)
Examples: (123) · (345) = (12345); (12) · (34) · (23) = (1234) . A more sophisticated computation:(12) · (12) = (12) · (cid:0) (13) + (24) − (34) (cid:1) = (123) + (124) − (12) · (34) . Proposition 19.
A cup product of perfect cycles is a perfect cycle.Proof.
Clearly, each perfect cycle is a product of elementary ones. Let us prove that the product of twoperfect cycles is an integer linear combination of perfect cycles. We may assume that the second factor isan elementary perfect cycle, say, (12). Let the first factor be ( A ) · ( A ) · ( A ) · . . . · ( A k ).We need the following case analysis: That is do not intersect the initial space M ,n . OMPACTIFICATIONS OF M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES 9 (1) If at least one of 1 , S A i , the product is a perfect cycle by Proposition 18,(1).(2) If 1 and 2 belong to different A i , we use the following:for any perfect cycle ( A ) · ( A ) with i ∈ A , j ∈ A , we have( A ) · ( A ) ⌣ ( ij ) = ( A ) ⌣ ( ij ) ⌣ ( A ) = ( A j ) ⌣ ( A ) = ( A ∪ A ) . (3) Finally, assume that 1 , ∈ A . Choose i / ∈ A , j / ∈ A such that i and j do not belong to oneand the same A k . By Proposition 18, (3),( A ) · ( A ) · ( A ) · . . . · ( A k ) ⌣ (12) = ( A ) · ( A ) · ( A ) · . . . · ( A k ) ⌣ (cid:0) (1 i ) + (2 j ) − ( ij ) (cid:1) . After expanding the brackets, one reduces this to the above cases. (cid:3)
Lemma 20.
For an ASD complex K , let A ⊔ B ⊔ C = [ n ] be a partition of [ n ] into three faces. Then ( A ) · ( B ) · ( C ) = 1 in the graded component A n − K of the Chow ring, which is canonically identified with Z .Proof. Indeed, the cycles ( A ), ( B ), and ( C ) intersect transversally at a unique point. (cid:3) Now we see that the set of perfect cycles is closed under cup-product. In the next section we show thatthe Chow ring equals the ring of perfect cycles.2.5.
Flips and blow ups.
Let K be an ASD complex, and let A ⊂ [ n ] be its facet. Lemma 21.
The perfect cycle ( A ) is isomorphic to M , P | Ac | +1 ∼ = P | A c |− .Proof. Contraction of A gives the complex pt ⊔ ∆ | A c | = P | A c | +1 from the Example 16, (2). (cid:3) Lemma 22.
For an ASD complex K and its facet A , there are two blow up morphisms π A : M ,K \ A → M ,K and π A c : M ,K \ A → M , flip A ( K ) . The centers of these blow ups are the perfect cycles ( A ) and ( A c ) respectively. The exceptional divisorsare equal: D A = D A c . Both are isomorphic to M , P | A | +1 × M , P | Ac | +1 ∼ = P | A | × P | A c | . The maps π A | D A and π A c | D Ac are projections to the first and the second components respectively. The proof literally repeats [8, Corollary 3.5]: K –stable but not K A ( K A c )–stable curves have twoconnected components. The marked points with indices from the set A lie on one of the irreduciblecomponents, and marked points with indices from the set A c lie on the other. (cid:3) Corollary 23.
For an ASD complex K and its facet A , the algebraic varieties M ,K and M ,K \ A areHI–schemes, i.e., the canonical map from the Chow ring to the cohomology ring is an isomorphism.Proof. This follows from Lemma 22 and Theorem 41. (cid:3) Chow rings of ASD compactifications
As it was already mentioned, many examples of ASD compactifications are polygon spaces, that is,come from a threshold ASD complex. Their Chow rings were computed in [2]. A more relevant to thepresent paper presentation of the ring is given in [3]. We recall it below.
Definition 24.
Let A ∗ univ = A ∗ univ,n be the ring Z (cid:2) ( I ) : I ⊂ [ n ] , ≤ | I | ≤ n − (cid:3) factorized by relations:(1) “ The four-term relations ”: ( ij ) + ( kl ) − ( ik ) − ( jl ) = 0 for any i, j, k, l ∈ [ n ].(2) “ The multiplication rule ”: ( Ik ) · ( J k ) = (
IJ k ) for any disjoint
I, J ⊂ [ n ] not containing element k .There is a natural graded ring homomorphism from A ∗ univ to the Chow ring of an ASD-compactificationthat sends each of the generators ( I ) to the corresponding perfect cycle. M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES
Theorem 25. [3] The Chow ring (it equals the cohomology ring) of a polygon space equals the ring A ∗ univ factorized by ( I ) = 0 whenever I is a long set. The following generalization of Theorem 25 is the first main result of the paper:
Theorem 26.
For an ASD complex L , the Chow ring A ∗ L := A ∗ ( M ,L ) of the moduli space M ,L isisomorphic to the quotient A ∗ univ by the ideal I L := (cid:10) ( I ) : I L (cid:11) . The idea of the proof is: the claim is true for threshold ASD complexes (i.e., for polygon spaces), andeach ASD complex is achievable from a threshold ASD complex by a sequence of flips. Therefore it issufficient to look at a unique flip. Let us consider an ASD complex K + B where B / ∈ K is a facet in K + B . Set A := [ n ] \ B , and consider the ASD complex K + A = flip B ( K + B ). KK + B K + A flip B We are going to prove that if the claim of the theorem holds true for K + B , then it also holds for K + A .By Lemma 22, the space M ,K is the blow up of M ,K + B along the subvariety ( B ) and the blow upof M ,K + A along the subvariety ( A ). The diagram of the blow ups looks as follows:( B ) D ( A ) M ,K + B M ,K M ,K + Ai B j A = j B g A g B i A π A π B The induced diagram of Chow rings is: A ∗ ( B ) = A ∗P | A | +1 A ∗P | A | +1 × A ∗P | B | +1 A ∗ ( A ) = A ∗P | B | +1 A ∗ K + B A ∗ K A ∗ K + Ag ∗ B g ∗ A i ∗ B π ∗ B j ∗ A = j ∗ B π ∗ A i ∗ A Let A ∗ K + A,comb be the quotient of A ∗ univ by the ideal I K + A . We have a natural graded ring homomor-phism α = α K + A : A ∗ K + A,comb → A ∗ K + A =: A ∗ K + A,alg , where the map α sends each symbol ( I ) to the associated perfect cycle.A remark on notation: as a general rule, all objects related to A ∗ K + A,comb we mark with a subscript “comb” , and objects related to A ∗ K + A,alg we mark with “alg” .We shall show that α is an isomorphism. The outline of the proof is:(1) The ring A ∗ K + A,alg is generated by the first graded component. (The ring A ∗ K + A,comb is alsogenerated by the first graded component; this is clear by construction.)(2) The restriction of α to the first graded components is a group isomorphism. Therefore, α issurjective.(3) The map α is injective. Lemma 27.
The ring A ∗ K + A,alg is generated by the group A K + A,alg .Proof.
By Theorem 38 A ∗ K ∼ = A ∗ K + A,alg [ T ] (cid:0) f A ( T ) , T · ker( i ∗ A ) (cid:1) . Observe that:
OMPACTIFICATIONS OF M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES 11 • The zero graded components of A ∗ K + A,alg , A ∗ K + A,comb equals Z . • The map π ∗ A : A ∗ K + A,alg → A ∗ K is a homomorphism of graded rings. Moreover, the variable T stands for the additive inverse of the class of the exceptional divisor D . And so, T a degree onehomogeneous element. • Since i ∗ A is the multiplication by the cycle ( A ), the kernel ker( i ∗ A ) equals the annihilator Ann( A ) alg in the ring A ∗ K + A,alg . Since the space M ,K + A is an HI-scheme, the degree of the ideal Ann( A ) alg is strictly positive. • The polynomial f A ( T ) is a homogeneous element whose degree equals the degree deg T ( f A ( T )).Besides, its coefficients are generated by elements from the first graded component since they allbelong to the ring α ( A ∗ K + A,comb ).Denote by h A K + A,alg i the subalgebra of A K + A,alg generated by the first graded component.First observe that the restriction of the map A ∗ K + A,alg [ T ] → A ∗ K to the first graded components isinjective.Assuming that the lemma is not true, consider a homogeneous element r of the ring A ∗ K + A,alg withminimal degree among all not belonging to h A K + A,alg i . There exist elements b i ∈ h A K + A,alg i such that b p · T p + · · · + b · T + b = r in the ring A ∗ K + A,alg [ T ]. The elements b i are necessarily homogeneous.Equivalently, b p · T p + · · · + b · T + b − r belongs to the ideal (cid:0) f A ( T ) , T · Ann( A alg ) (cid:1) . Therefore b p · T p + · · · + b · T + b − r = x · f A ( T ) + y · T · i with some x, y ∈ R [ T ] and i ∈ Ann( A alg ).Setting T = 0, we get b − r = x · f . If the element x belongs to h A K + A,alg i , then we are done.Assume the contrary. Then from the minimality assumption we get the following inequalities: deg( b − r ) =deg( x · f ) > deg( x ) ≥ deg( r ). A contradiction. (cid:3) Lemma 28.
For any ASD complex L the groups A L,comb and A L,alg are isomorphic. The isomorphismis induced by the homomorphism α L . The proof analyses how do these groups change under flips.We know that the claim is true for threshold complexes. Due to Lemma 10 we may consider flips onlywith n − > | A | >
2. Again, we suppose that the claim is true for the complex K + B and will prove forthe complex K + A with A ⊔ B = [ n ]. Under such flips A comb does not change. The group A does notchange neither. This becomes clear with the following two short exact sequences (see Theorem 39,e):0 → A n − (cid:0) M , P | A | +1 (cid:1) → A n − (cid:0) M , P | A | +1 × M , P | B | +1 (cid:1) ⊕ A n − (cid:0) M ,K + B (cid:1) → A n − (cid:0) M ,K (cid:1) → , → A n − (cid:0) M , P | B | +1 (cid:1) → A n − (cid:0) M , P | A | +1 × M , P | B | +1 (cid:1) ⊕ A n − (cid:0) M ,K + A (cid:1) → A n − (cid:0) M ,K (cid:1) → . (cid:3) Now we know that α : A ∗ K + A,comb → A ∗ K + A,alg is surjective.
Proposition 29.
Let Γ be a graph V ert (Γ) = [ n ] which equals a tree with one extra edge. Assume thatthe unique cycle in Γ has the odd length. Then the set of perfect cycles { ( ij ) } corresponding to the edgesof Γ is a basis of the (free abelian) group A univ .Proof. Any element of the group A univ by definition has a form P ij a ij · ( ij ) with the sum ranges overall edges of the complete graph on the set [ n ]. The four-term relation can be viewed as an alternatingrelation for a four-edge cycle. One concludes that analogous alternating relation holds for each cycle ofeven length. Example: ( ij ) − ( jk ) + ( kl ) − ( lm ) + ( mp ) − ( pi ) = 0. Such a cycle may have repeatingvertices. Therefore, if a graph has an even cycle, the perfect cycles associated to its edges are dependant.It remains to observe that the graph Γ is a maximal graph without even cycles. (cid:3) By Theorem 38, the Chow rings of the compactifications corresponding to complexes K , K + A , and K + B are related in the following way: A ∗ K ∼ = A ∗ K + A,alg [ T ] (cid:0) f A ( T ) , T · ker( i ∗ A ) (cid:1) ∼ = A ∗ K + B [ S ] (cid:0) f B ( S ) , S · ker( i ∗ B ) (cid:1) . Now we need an explicit description of the polynomials f A and f B . M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES x=xxxx a . . . y=yyyy b . . . Figure 3. A = { x, x , . . . , x a } B = { y, y , . . . , y b } , where | A | = a and | B | = b , take the generators (cid:8) ( xy ); ( xy i ) , i ∈ { , . . . , b } ; ( x j y ) , j ∈ { , . . . , a } ; ( yy ) (cid:9) for A ∗ K + B , and (cid:8) ( xy ); ( xy i ) , i ∈ { , . . . , b } ; ( x j y ) , j ∈ { , . . . , a } ; ( xx ) (cid:9) for A ∗ K + A,comb . Denote by A the subring of the Chow rings A ∗ K + A,comb and A ∗ K + B generated by the elements { ( xy ); ( xy i ) , i ∈{ , . . . , b − } ; ( x j y ) , j ∈ { , . . . , a − }} .Then A ∗ K + A,comb is isomorphic to A [ I ] /F B ( I ) where I := ( xx ) and F B ( I ) is an incarnation of theexpression ( B ) = ( yy ) · · · · · ( yy b ) = 0 via the generators. Analogously, A ∗ K + B,comb ∼ = A [ J ] /F A ( J ) with V := ( yy ).The cycles ( A ) and ( B ) equal to the complete intersection of divisors ( xx ) , ( xx ) , . . . , ( xx a − ) and( yy ) , ( yy ) , . . . , ( yy b − ) respectively. So the Chern polynomials are: f A ( T ) = (cid:0) T + ( xx ) (cid:1) · · · · · (cid:0) T + ( xx a − ) (cid:1) and f B ( S ) = (cid:0) S + ( yy ) (cid:1) · · · · · (cid:0) S + ( yy b − ) (cid:1) . Moreover, the new variables T and S correspond to one and the same exceptional divisor D A = D B .Relation between polynomials f • and F • are clarified in the following lemma. Lemma 30.
The Chow class of the image of a divisor ( ab ) K + A , a, b ∈ [ n ] under the morphism π ∗ A equals (1) ( ab ) K for a ∈ A, b ∈ B , or vice versa; (2) bl ( ab )( A ) (cid:0) ( ab ) K + A (cid:1) for { a, b } ⊂ B ; (3) bl ( A ) (cid:0) ( ab ) K + A (cid:1) + D A for { a, b } ⊂ A .Proof. In case (1), the cycle ( ab ) K + A does not intersect ( A ) K + A . It is by definition bl ( ab ) ∩ ( A ) (cid:0) ( ab ) K + A (cid:1) .Then (1) and (2) follow directly from Theorem 40,(2) by dimension counts.The claim (3) also follows from the blow-up formula Theorem 40: π ∗ A ( ab ) = bl ( ab ) ∩ ( A ) (cid:0) ( ab ) K + A (cid:1) + j A, ∗ (cid:0) g ∗ A [( ab ) ∩ ( A )] · s ( N D A M ,K ) (cid:1) n − , where N D A M ,K is a normal bundle and s ( ) is a total Segre class. This follows from the equalities bythe functoriality of the total Chern and Segre classes and the equality s ( U ) · c ( U ) = 1. Namely, we have s (( ab ) ∩ ( A )) = [( ab ) ∩ ( A )] · s ( N ( ab ) ∩ ( A ) ( ab ) K + A ) = [( ab ) ∩ ( A )] · s (cid:0) N ( A ) M ,K + A (cid:1) ,c g ∗ A N ( A ) M ,K + A N D A M ,K ! · g ∗ A [( ab ) ∩ ( A )] · g ∗ A s (cid:0) N ( A ) M ,K + A (cid:1) = g ∗ A [( ab ) ∩ ( A )] · s ( N D A M ,K ) . Finally, we note that g ∗ A [( ab ) ∩ ( A )] = D A . (cid:3) From Lemma 30, we have the equality f A ( T ) = π ∗ A (cid:0) ( xx ) · · · · · ( xx a − ) (cid:1) = π ∗ A ( A ) and f B ( S ) = π ∗ B ( B ) . OMPACTIFICATIONS OF M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES 13
Lemma 31. (1)
The ideal
Ann( A ) comb is generated by its first graded component. (2) More precisely, the generators of the ideal
Ann( A ) comb are (a) the elements of type ( ab ) with a ∈ A, b ∈ B , and (b) the elements of type ( a a ) − ( b b ) , where a , a ∈ A ; b , b ∈ B = A c . (3) The annihilators
Ann( A ) comb and Ann( B ) are canonically isomorphic.Proof. First observe that the kernel ker( i ∗ A ) equal the annihilator of the cycle ( A ). Set κ be the idealgenerated by ker( i ∗ A ) ∩ A . Without loss of generality, we may assume that A = { , , ..., m } . Observethat:(1) If I ⊂ [ n ] has a nonempty intersection with both A and A c , then ( A )( I ) = 0. In this case, ( I ) canbe expressed as ( ab )( I ′ ), where a ∈ A, b ∈ A c .(2) (i) If I ⊂ A , then ( A ) ⌣ ( I ) = ( A )( m + 1 ...m + | I | ) . (ii) In this case, the element ( I ) − ( m + 1 , ..., m + | I | ) is in κ .Let us demonstrate this by giving an example with A = { , , , } , ( I ) = (12):(1234) ⌣ (12) = ( A ) ⌣ ((15)+ (16) − (56)) = 0+ 0 − (1234)(56). We conclude that (12) − (56) ∈ κ .Let us show that (123) − (567) ∈ κ .Indeed, (123) − (567) = (12) ⌣ (23) − (567) ∈ κ ⇔ (56) ⌣ (23) − (567) ∈ κ ⇔ (56) ⌣ ((23) − (67)) ∈ κ . Since (23) − (67) ∈ κ , the claim is proven.(3) If I ⊂ A c , then ( A ) ⌣ ( I ) = ( A )( m + 1 , ..., m + | I | ) . The element ( I ) − ( m + 1 , ..., m + | I | ) is in κ .This follows from (1) and (2).Now let us prove the lemma. Assume x ⌣ ( A ) = 0. Let x = P i a i ( I i ) ... ( I ik i ) . We may assume that x is a homogeneous element. Modulo κ , each summand ( I ) ... ( I k i ) can be reducedto some ( m + 1 ...m + r )( m + r + 1 , ..., m + r ) ... ( m + r k i + 1 ...m ′ ). Modulo κ , ( m + 1 ...m + r )( m + r + 1 ...m + r ) ... ( m + r k i + 1 , ..., m ′ ) can be reduced to a one-bracket element ( m + 1 ...m + r m + r +1 ...m + r ...m + r k i + 1 , ..., m ′′ ).Indeed, for two brackets we have:( m +1 ...m + r )( m + r +1 , ..., m + r ) ≡ ( m +1 ...m + r )( m + r , ..., m + r − ≡ ( m +1 ...m + r −
1) (mod κ ) . For a bigger number of brackets, the statement follows by induction.We conclude that a homogeneous x ∈ ker( i ∗ A ) modulo κ reduces to some a ( m +1 ...m + m ′ ), where a ∈ Z .Then a = 0. Indeed, ( A )( m + 1 ...m + m ′ ) = 0 since by Lemma 20 ( A )( m + 1 ...m + m ′ )( m + m ′ ...n ) = 0. (cid:3) Remark 32.
Via the four-term relation any element from b) can be expressed as a linear combinationof elements from a). So only a)–elements are sufficient to generate the annihilators. Actually,
A ∼ = C ⊕ Ann( A ) comb . We arrive at the following commutative diagram of graded rings: A ∗ K ∼ = g A ∗ K / Ann( B ) ∼ = g A ∗ K / Ann( A ) comb g A ∗ K := A ∗ K + A,comb [ T ] /f ( A ) ( T ) ∼ = A ∗ K + B [ S ] /f ( B ) ( S ) A ∗ K + B ∼ = A [ J ] /F A ( J ) A ∗ K + A,comb ∼ = A [ I ] /F B ( I ) A f B = π ∗ B F B f A = π ∗ A F A F A F B M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES
Therefore, the following diagram commutes:0 Ann( A ) comb A ∗ K + A,comb [ f ( A ) ] A ∗ K + A,comb [ f ( A ) ] / Ann( A ) comb
00 Ann( A ) alg A ∗ K + A,alg [ f ( A ) ] A ∗ K + A,alg [ f ( A ) ] / Ann( A ) alg α ∼ = Here R [ g ] denotes the extension of a ring R by a polynomial g ( t ). All three vertical maps are induced bythe map α ; the last vertical map is an isomorphism since both rings are isomorphic to A ∗ K .The ideals Ann( A ) comb and Ann( A ) alg coincide, so the homomorphism α : A ∗ K + A,comb [ f ( A ) ] → A ∗ K + A,alg [ f ( A ) ]is injective, and the theorem is proven. (cid:3) Poincar´e polynomials of ASD compactifications
Theorem 33.
Poincar´e polynomial P q (cid:0) M ,L (cid:1) for an ASD complex L equals P q (cid:0) M ,L (cid:1) = 1 q ( q − (1 + q ) n − − X I ∈ L q | I | ! . Proof.
This theorem is proven by Klyachko [10, Theorem 2.2.4] for polygon spaces, that is, for compacti-fications coming from a threshold ASD complex. Assume that K + A be a threshold ASD complex. Forthe blow up of the space M ,K + B along the subvariety ( B ) we have an exact sequence of Chow groups0 → A p (cid:0) M , P | A | +1 (cid:1) → A p (cid:0) M , P | A | +1 × M , P | B | +1 (cid:1) ⊕ A p (cid:0) M ,K + B (cid:1) → A p (cid:0) M ,K (cid:1) → , where, as before, A ⊔ B = [ n ] and p is a natural number. We get the equality P q ( K ) = P q ( K + B ) + P q ( P | A | +1 ) · P q ( P | B | +1 ) − P q ( P | A | +1 ) . Also, we have the recurrent relations for the Poincar´e polynomials: P q ( K + B ) = P q ( K + A ) + P q ( P | A | +1 ) − P q ( P | B | +1 ) == 1 q ( q − (1 + q ) n − − X I ∈ K + A q | I | + q | A | − q − q | B | + q ! == 1 q ( q − (1 + q ) n − − X I ∈ K q | I | − q | A | + q | A | − q − q | B | + q ! == 1 q ( q − (1 + q ) n − − X I ∈ K + B q | I | ! . We have used the following: if U is a facet of an ASD complex, then there is an isomorphism M , P | U | +1 ∼ = P | U |− (see Example 16(2) ); the Poincar´e polynomial of the projective space P | U |− equals q | U | − qq ( q − . (cid:3) The tautological line bundles over M ,K and the ψ –classes The tautological line bundles L i , i = 1 , ..., n were introduced by M. Kontsevich [4] for the Deligne-Mumford compactification. The first Chern classes of L i are called the ψ - classes .We now mimic the Kontsevich’s original definition for ASD compactifications. Let us fix an ASDcomplex K and the corresponding compactification M ,K . Definition 34.
The line bundle E i = E i ( L ) is the complex line bundle over the space M ,K whose fiberover a point ( u , ..., u n ) ∈ ( P ) n is the tangent line to the projective line P at the point u i . The firstChern class of E i is called the ψ -class and is denoted by ψ i . In the original Kontsevich’s definition, the fiber over a point is the cotangent line, whereas we have the tangent line.This replacement does not create much difference.
OMPACTIFICATIONS OF M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES 15
Proposition 35. (1)
For any i = j = k ∈ [ n ] we have ψ i = ( ij ) + ( ik ) − ( jk ) . (2) The four-term relation holds true: ( ij ) + ( kl ) = ( ik ) + ( jl ) for any distinct i, j, k, l ∈ [ n ] . Proof. (1) Take a stable configuration ( x , ..., x n ) ∈ M ,K . Take the circle passing through x i , x j , and x k .It is oriented by the order ijk . Take the vector lying in the tangent complex line to x i which is tangentto the circle and points in the direction of x j . It gives rise to a section of E i which is defined correctlywhenever the points x i , x j , and x k are distinct. Therefore, ψ i = A ( ij ) + B ( ik ) + C ( jk ) for some integer A, B, C . Detailed analysis specifies their values.Now (2) follows since the Chern class ψ i does not depend on the choice of j and k . (cid:3) Let us denote by | d , . . . , d n | K the intersection number h ψ d ...ψ d k k i K = ψ ⌣d ⌣ ... ⌣ ψ ⌣d k k related tothe ASD complex K . Theorem 36.
Let M ,K be an ASD compactification. A recursion for the intersection numbers is | d , . . . , d n | K = | d , . . . , d i + d j − , . . . , ˆ d j , . . . , d n | K ( ij ) + | d , . . . , d i + d k − , . . . , ˆ d k , . . . , d n | K ( ik ) − | d , . . . , d i − , . . . , d j + d k , . . . , ˆ d j , ˆ d k , . . . , d n | K ( jk ) , where i, j, k ∈ [ n ] are distinct.Remind that K ( ij ) denotes the complex K with i and j frozen together. Might happen that K ( ij ) isill-defined, that is, ( ij ) / ∈ K . Then we set the corresponding summand to be zero.Proof. By Proposition 35, h ψ d . . . ψ d n n i K = h ψ d − . . . ψ d n n i K ⌣ (cid:0) (1 i ) + (1 j ) − ( ij ) (cid:1) . It remains to observe that h ψ d − . . . ψ d n n i K ⌣ ( ab ) equals the h ψ d − . . . ψ d n n i K ( a,b ) . (cid:3) Theorem 37.
Let M ,K be an ASD compactification. Any top monomial in ψ -classes modulo renumber-ing has a form ψ d ⌣ ... ⌣ ψ d m m with P mq =1 d q = n − and d q = 0 for q = 1 , ..., m . Its value equals the signed number of partitions [ n −
2] = I ∪ J with m + 1 ∈ I and I, J ⊂ K . Each partition is counted with the sign ( − N · ε, where N = | J | + X q ∈ J,q ≤ m d q , ε = , if J ∪ { n } ∈ K, and J ∪ { n − } ∈ K ; − , if I ∪ { n } ∈ K, and I ∪ { n − } ∈ K ;0 , otherwise.Proof goes by induction.Although the base is trivial, let us look at it. The smallest n which makes sense is n = 4. There existtwo ASD complexes with four vertices, both are threshold. So there exist two types of fine moduli com-pactifications, both correspond to the configuration spaces of some flexible four-gon. The top monomialsare the first powers of the ψ -classes.(1) For l = 1; l = 1; l = 1; l = 0 , ψ = ψ = ψ = 0, and ψ = 2.Let us prove that the theorem holds for the monomial ψ . There are two partitions of[ n −
2] = [2]:(a) J = { } , I = { } . Here ε = 0, so this partition contributes 0.(b) J = ∅ , I = { , } . Here I / ∈ K , so this partition also contributes 0. M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES (2) For l = 2 , l = 1; l = 1; l = 1, we have ψ = ψ = ψ = 1, and ψ = −
1. Let us checkthat the theorem holds for the monomial ψ . (The other monomials are checked in a similar way.)There partitions of [2] are the same:(a) J = { } , I = { } . Here ε = − , N = 1 + 1, so this partition contributes − J = ∅ , I = { , } . Here I / ∈ K , so it contributes 0.For the induction step , let us use the recursion. We shall show that for any partition [ n −
2] = I ∪ J ,its contribution to the left hand side and the right hand side of the recursion are equal.This is done through a case analysis. We present here three cases; the rest are analogous.(1) Assume that i, j, k ∈ I , and ( I, J ) contributes 1 to the left hand side count. Then ⊲ ( d , . . . , d i + d j − , . . . , ˆ d j , . . . , d n ) K ( ij ) contributes 1 to the right hand side.Indeed, neither N , nor ε changes when we pass from K to K ( ij ) . ⊲ ( d , . . . , d i + d k − , . . . , ˆ d k , . . . , d n ) K ( ik ) contributes 1, and ⊲ − ( d , . . . , d i − , . . . , d j + d k , . . . , ˆ d j , ˆ d k , . . . , d n ) K ( jk ) contributes − i ∈ I, j, k ∈ J , and ( I, J ) contributes 1 to the left hand side count. Then ⊲ ( d , . . . , d i + d j − , . . . , ˆ d j , . . . , d n ) K ( ij ) contributes 0 to the right hand side. ⊲ ( d , . . . , d i + d k − , . . . , ˆ d k , . . . , d n ) K ( ik ) contributes 0, and ⊲ − ( d , . . . , d i − , . . . , d j + d k , . . . , ˆ d j , ˆ d k , . . . , d n ) K ( jk ) contributes 1.Indeed, N turns to N −
1, whereas ε stays the same.(3) Assume that i ∈ J, j, k ∈ I , and ( I, J ) contributes 1 to the left hand side count. Then ⊲ ( d , . . . , d i + d j − , . . . , ˆ d j , . . . , d n ) K ( ij ) contributes 0. ⊲ ( d , . . . , d i + d k − , . . . , ˆ d k , . . . , d n ) K ( ik ) contributes 0, and ⊲ − ( d , . . . , d i − , . . . , d j + d k , . . . , ˆ d j , ˆ d k , . . . , d n ) K ( jk ) contributes 1, since N turns to N − ε stays the same. (cid:3) This theorem was proven for polygon spaces (that is, for threshold ASD complexes) in [14].6.
Appendix. Chow rings and blow ups
Assume we have a diagram of a blow up e Y := bl X ( Y ). Here X and Y are smooth varieties, ι : X ֒ → Y is a regular embedding, and e X is the exceptional divisor. In this case, ι ∗ : A ∗ ( Y ) → A ∗ ( X ) is surjective. e X e YX Y τ θ πι
Denote by E the relative normal bundle E := τ ∗ N X Y /N e X e Y .
Theorem 38. [7, Appendix. Theorem 1] The Chow ring A ∗ ( e Y ) is isomorphic to A ∗ ( Y )[ T ]( P ( T ) , T · ker i ∗ ) , where P ( T ) ∈ A ∗ ( Y )[ T ] is the pullback from A ∗ ( X )[ T ] of Chern polynomial of the normal bundle N X Y .This isomorphism is induced by π ∗ : A ∗ ( Y )[ T ] → A ∗ ( e Y ) which sends − T to the class of the exceptionaldivisor e X . Theorem 39. [15, Proposition 6.7] Let k ∈ N . a) (Key formula) For all x ∈ A k ( X ) π ∗ ι ∗ ( x ) = θ ∗ ( c d − ( E ) ∩ τ ∗ x ) OMPACTIFICATIONS OF M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES 17 e) There are split exact sequences → A k X υ −→ A k e X ⊕ A k Y η −→ A k e Y → with υ ( x ) = (cid:0) c d − ( E ) ∩ τ ∗ x, − ι ∗ x (cid:1) , and η (˜ x, y ) = θ ∗ ˜ x + π ∗ y . A left inverse for υ is given by (˜ x, y ) τ ∗ (˜ x ) . Theorem 40. [15, Theorem 6.7, Corollary 6.7.2] (1) (Blow-up Formula) Let V be a k -dimensional subvariety of Y , and let ˜ V ⊂ ˜ Y be the propertransform of V , i.e., the blow-up of V along V ∩ X . Then π ∗ [ V ] = [ ˜ V ] + j ∗ { c ( E ) ∩ τ ∗ s ( V ∩ X, V ) } k in A k ˜ Y . (2) If dim V ∩ X ≤ k − d , then π ∗ [ V ] = [ ˜ V ] . An algebraic variety Z is a HI–scheme if the canonical map cl : A ∗ ( Z ) → H ∗ ( Z, Z ) is an isomorphism. Theorem 41. [7, Appendix. Theorem 2] • If X, e X , and Y are HI, then so is e Y . • If X, e X , and e Y are HI, then so is Y . Acknowledgement.
This research is supported by the Russian Science Foundation under grant 16-11-10039.
References [1] Smyth, D.I. Invent. math. (2013) 192: 459. https://doi.org/10.1007/s00222-012-0416-1 .[2] J.-C. Hausmann, A. Knutson,
The cohomology ring of polygon spaces , Annales de l’institut Fourier48, 1 (1998), 281–321.[3] I. Nekrasov, and G. Panina. Geometric presentation for the cohomology ring of polygon spaces,arXiv:1801.00785.[4] M. Kontsevich
Intersection theory on the moduli space of curves and the matrix Airy function , Comm.Math. Phys., 147, 1(1992), 1–23.[5] P. Galashin, G. Panina
Manifolds associated to simple games , J. Knot Theory Ramifications,25(12):1642003, 14, 2016.[6] Deligne, P. & Mumford, D. Publications Math´ematiques de l’Institut des Hautes Scientifiques (1969)36: 75–109. https://doi.org/10.1007/BF02684599 .[7] Keel, Sean. “Intersection Theory of Moduli Space of Stable N-Pointed Curves of Genus Zero.” Vol.330, no. 2, 1992, pp. 545–574., https://doi.org/10.2307/2153922 .[8] Hassett, Brendan. Moduli spaces of weighted pointed stable curves, Advancesin Mathematics, Volume 173, Issue 2, 2003, Pages 316-352, ISSN 0001-8708, https://doi.org/10.1016/S0001-8708(02)00058-0 .[9] Ceyhan, ¨Ozg¨ur. Chow groups of the moduli spaces of weighted pointed stable curves of genuszero, Advances in Mathematics, Volume 221, Issue 6, 2009, Pages 1964-1978, ISSN 0001-8708, https://doi.org/10.1016/j.aim.2009.03.011 .[10] A. Klyachko,
Spatial polygons and stable configurations of points in the projective line , In: TikhomirovA., Tyurin A. (eds) Algebraic Geometry and its Applications. Aspects of Mathematics, vol 25 (1994),67-84.[11] Von Neumann, J., Morgenstern, O.: Theory of games and economic behavior. Bull. Amer. Math. Soc51, 498-504 (1945)[12] H.-B. Moon, C. Summers, J. von Albade, and R. Xie. Birational contractions of ¯ M ,n and combina-torics of extremal assignments. J. Algebraic Comb., Vol. 47, (2018), no. 1, 51–90.[13] A. Losev and Y. Manin. New moduli spaces of pointed curves and pencils of flat connections, MichiganMath. J., Volume 48, Issue 1 (2000), 443-472. https://doi.org/10.1307/mmj/1030132728 . M ,n ASSOCIATED WITH ALEXANDER SELF DUAL COMPLEXES [14] J. Agapito, L. Godinho,