Intersections of $ψ$ classes on Hassett Spaces for genus $0$ with all weights $\frac{1}{2}$
Nand Sharma
math.AG math.CO






Department of Mathematics, Colorado State University, CO, USA
sharma@math.colostate.edu

[2000]14Q99


Hassett spaces are moduli spaces of weighted stable pointed curves. In this work, we consider such spaces of curves of genus 0 with weights all 1/2. These spaces are interesting as they are isomorphic to M_0,n but have different universal families and different intersection theory. We develop a closed formula for  intersections of ψ-classes on such spaces. In our main result, we encode the formula for top intersections in a generating function obtained by applying a differential operator to the Witten-potential. 


ψ class intersections on Hassett Spaces for genus 0 with all weights 1/2
    Nand Sharma
    December 30, 2023
========================================================================





§ INTRODUCTION

The moduli space of algebraic curves of genus 0 with n marked points, M_0, n (with the Deligne-Mumford compactification <cit.>) has been an important topic of research in algebraic geometry. These spaces provide an algebro-geometric tool to study how pointed rational curves vary in families, and are of fundamental importance in areas like Gromov-Witten theory and topological quantum field theories <cit.>.

In <cit.>, Hassett constructed a new class of modular compactifications M_0, 𝒜 of the moduli space M_0, n of smooth curves with n marked points parameterized by an input datum 𝒜, consisting of a collection 𝒜 = (a_1 , … , a_n ) of weights a_i ∈ Q ∩ (0, 1] such that a_1 + … + a_n > 2. We call these spaces M_0, 𝒜 the Hassett spaces of rational curves.

A lot of work is being done on Hassett spaces including developing their tautological intersection theory and weighted Gromov-Witten theory, e.g. in <cit.>, <cit.> and <cit.>.In this work, we contribute to the tautological intersection theory of a special case of such spaces- Hassett spaces of rational curves with weights all 1/2, denoted M_0,( 1/2)^n. These spaces provide for interesting spaces for combinatorial results in its intersection theory because of the symmetry of weights and its connections with intersection theory for M_0,n.These spaces are interesting also because they are fine moduli spaces, are isomorphic to M_0,n, but have different universal families and different intersection theory. Exploring these differences and developing some results in its tautological intersection theory is the contribution of this work.
For this work, the following notations are used: a ψ class on M_0,n is denoted as ψ_i; a ψ class on M_0,( 1/2)^n is denoted as ψ̅_i, and the pullback of a ψ class under the reduction morphism from M_0,n to M_0,( 1/2)^n is denoted ψ̂_i.
     
In our first result <ref>, we develop a closed formula (<ref>) for the monomials in ψ̂ classes in terms of cycles on M_0,n. This closed formula is derived using the relation (<ref>) between the ψ̂ classes and ψ classes on M_0,n, in which ψ_i is corrected by all boundary divisors where the i-th mark is on a twig that gets contracted when pushed forward to M_0,( 1/2)^n. The proof uses this relation to obtain the ψ̂ monomials as monomials in ψ classes and boundary divisors on M_0,n. So, the summands in the resulting expansion correspond to modified ψ monomials on certain boundary strata on M_0,n that are the intersections of these boundary divisors.The dual graphs of these strata are all `forked' graphs - graphs with a `central' node and some `forks', e.g. figure (<ref>).We then establish a bijection between summands in the expansion corresponding to these graphs that we call `𝒫'-graphs and the unordered partitions of [n], such that cardinality of each subset in the partition is either 2 or 1. The resulting formula (<ref>) has the pullback of monomials in ψ̅ classes on M_0,( 1/2)^n as a sum of the intersections of monomials in ψ classes and boundary strata corresponding to 𝒫-graphs on M_0,n. Then we derive two corollaries (<ref> and <ref>) of this result to calculate the top intersections. These give the top intersections of ψ̂ classes as a sum of top intersections of ψ classes on M_0,n, M_0,n-1, M_0,n-2, … with some multiplicities. We point out here that our corollaries (<ref> and <ref>) can also be deduced from theorem 7.9 in <cit.>. For our work, we develop specific and explicit closed formulas for our special case of all weights 1/2 and base our combinatorial analysis closely on the structure of dual graphs.

The main theorem <ref> of this work encodes the closed formula (<ref>) for top intersections in a generating function G(𝐭) obtained by applying a differential operator to the Witten-potential F(𝐭) <cit.>. This operator ℒ̂ takes the form of an exponential partial differential operator and provides a very nice compact way to describe these top intersections. 
For this, we define `𝒫_k-graphs' that are obtained by replacing the i-th mark with k_i on a 𝒫-graph, where k_i is the exponent of ψ̂_̂î in the ψ̂ monomial (<ref>). As expected, there is a surjection between 𝒫-graphs and 𝒫_k-graphs. Corresponding to these `𝒫_k-graphs', we write a new version of our closed formula in terms of these graphs (<ref>). Then we show a bijection between the summands in this formula and the summands in the coefficient of the appropriate term in ℒ̂( F(𝐭) ). The resulting coefficient, as a sum of all these summands, corresponds to the top intersections of ψ̂ classes.

The paper is organized as follows. In section <ref>, we give the background required for this work which consists of a brief introduction to M_0,n, ψ classes and Hassett Spaces, with some relevant theorems and lemmas on these topics. In section <ref>, we prove our first result which gives the closed formula for the intersections of ψ̂ classes. In section <ref>, we give the results for top intersections, and encode the formula for top intersections in the generating function that we obtain by applying a partial differential operator to the Witten-potential.




§ BACKGROUND



For background on this work, the author has mainly used <cit.>, <cit.>, <cit.>, <cit.> and introductory sections of <cit.>. Here we will recall some selected facts we explicitly use in this work.


 §.§ M_0,n

We denote by M_0,n the moduli space of stable, n pointed rational curves, with at worst nodal singularities. The boundary of M_0,n is defined to be the complement of M_0,n in M_0,n. It consists of all points parameterizing nodal stable curves.

Given a rational, stable n-pointed curve (C,p_1, …, p_n), its dual graph  is defined to have:


  * a vertex for each irreducible component of C;

  * an edge for each node of C, joining the appropriate vertices;

  * a labeled half edge for each marked point, emanating from the appropriate vertex.
 

Figure (<ref>) below gives an example of the dual graphs of some strata in M_0,5. 


The closures of the codimension 1 boundary strata of M_0,n are called the irreducible boundary divisors;
they are in one-to-one correspondence with all ways of
partitioning [n]=A∪ A^c with the cardinality of both A and  A^c strictly greater than 1. We denote D(A)= D(A^c) the divisor corresponding to the partition { A,A^c}. 



 §.§ ψ classes

For i= 1, …, n, we define  the class ψ_i ∈ A^1(M_0,n). 
Let L_i →M_0,n be a line bundle whose fiber over each point (C, p_1, …, p_n) is canonically identified with T^∗_p_i(C). The line bundle L_i is called the i-th cotangent (or tautological) line bundle. Then

    ψ_i := c_1(L_i )
 
where c_1 is the first Chern class of the line bundle L_i.

Some properties of ψ classes on M_0,n that we use are the following lemmas. The interested reader can find their proofs in <cit.>. Here [n]=I∪ I^c with the cardinality of both I and  I^c strictly greater than 1.

Consider  the gluing morphism gl_I: M_0, I ∪⋆×M_0, I^c ∪∙→M_0, n. Assume that i∈ I and denote by π_1:M_0, I ∪⋆×M_0, I^c ∪∙→M_0, I∪⋆ the first projection.
Then:

    gl_I^∗ (ψ_i) =  π_1^∗(ψ_i).



Consider the forgetful morphism π_n+1: M_0,n+1→M_0,n. Then, for every i = 1, …, n,

    ψ_i  = π_n+1^∗(ψ_i) + D({i, n+1}).



For any choice of i, j, k distinct, we have the following equation in A^1(M_0,n):

    ψ_i = ∑_i∈ I,   j,k ∉I D(I).



Consider the forgetful morphism π_n+1: M_0,n+1→M_0,n. Then

    π_n+1 ∗(  ∏_i=1^n ψ_i^k_i)=
    ∑_j |k_j≠0ψ_j^k_j-1∏_i≠jψ_i^k_i.



Let ∑ k_i = n-3. Then

    ∫_M_0,n∏_i=1^nψ_i^k_i = n-3k_1, …, k_n

where the integral sign denotes push-forward to the class of a point.




 §.§ Hassett spaces

In <cit.>, Hassett constructed a new class of modular compactifications M_g, 𝒜 of the moduli space M_g, n of smooth curves with n marked points parameterized by an input datum (g, 𝒜). Here g is the genus of the curves and 𝒜 = (a_1 , … , a_n ) is the weight data  of weights a_i ∈ Q ∩ (0, 1] satisfying the inequality 2g - 2 + a_1 + … + a_n > 0.

M_g, 𝒜 that we call Hassett space parameterizes curves (C, p_1 ,… , p_n ) with n marked non-singular points on C that are 𝒜-stable if the the following two conditions are fulfilled.


  * The twisted canonical divisor K_C + a_1 p_1 + … + a_n p_n is ample.

  * A subset p_i_1 , …, p_i_k of the marked points is allowed to coincide only if the inequality a_i_1 +…+ a_i_k≤ 1 holds.

For g=0, the stability condition means that a rational n-pointed  curve (C, p_1, …, p_n) is 𝒜-stable if on every irreducible component of C the number of nodes plus the sum of the weights of the marks lying on that component is strictly greater than 2; and on C, a_1 + … + a_n > 2. 
In the case (a_1 ,… , a_n ) = (1, … , 1), this condition is nothing but the traditional notion of an n-marked stable curve, and so the compactification M_g, 𝒜 is exactly the well-known
Deligne-Mumford compactification M_g, n of M_g, n in this case.

Given two weight data 𝒜, ℬ, we say that ℬ≤𝒜 if for every i, b_i≤ a_i. Then there exists a regular reduction morphism:

    c_ℬ,𝒜:M_0, 𝒜→M_0, ℬ

such that c_ℬ,𝒜(C,p_1, …, p_n) is obtained by contracting twigs that become unstable when the weights of the points are “lowered” from a_i to b_i.


Moduli spaces of weighted stable rational curves also have ψ classes, which are defined in the same way as for M_0,n. A ψ class on M_0, 𝒜 will be denoted as ψ̅_i.  

Consider the reduction morphism
c:  M_0,n→M_0,𝒜. For i = 1, …, n, we have:

    ψ_i = c^∗ (ψ̅_i) + ∑_i ∈ I,   a_I≤ 1 D(I),


where 
a_I = ∑_j∈ Ia_j.


Consider the following commutative diagram:

    0.6in1in_0,n[r]^C [d]_π   _0,𝒜[d]_π̅
    M_0,n[r]^c @/_/[u]_σ_i   M_0,𝒜@/_/[u]_σ̅_̅i̅

Here _0,n and _0,𝒜 are the universal families over M_0,n and M_0,𝒜 respectively; σ_i and σ̅_i are the i-th tautological sections of the corresponding universal families; and c and C are the reduction morphisms. 

Now, pulling back relative differentials, Ω_π and Ω_π̅,  yields a natural map

    C^∗Ω_π̅→Ω_π .

So we get an induced map

    σ_i^∗ C^∗Ω_π̅→σ_i^∗Ω_π.

Since C ∘σ_i = σ̅_i ∘ c, we may re-write the map above as

    p : c^∗σ̅_i^∗Ω_π̅→σ_i^∗Ω_π.

Note that both sides are line bundles on M_0,n, and their first Chern classes
are c^∗ψ̅_̅i̅ and ψ_i, respectively. An easy local calculation shows that the
map p is an isomorphism on the interior and has a simple zero along the boundary divisor D(I) if i ∈ I and a a_I ≤ 1. By comparing the first Chern
classes, we deduce

    ψ_i = c^∗ (ψ̅_i) + ∑_i ∈ I,   a_I≤ 1 D(I)

as asserted.
 
Informally, for the pullback of a ψ̅_̅i̅, a ψ_i is corrected by all boundary divisors where the i-th mark is on a twig that gets contracted by c. 

Define the ψ̂_̂î class as the pullback of a ψ̅ class under the reduction morphism c:  M_0,n→M_0,𝒜, 

    ψ̂_̂î := c^∗ψ̅_̅i̅.
 


For the reduction morphism
c:  M_0,n→M_0,𝒜, where 𝒜 = {1/2, 1/2, … ,1/2}, for i = 1, …, n, we have:

    ψ̂_i = ψ_i - ∑_j, j ≠ i D({  i,j}) .


 
This follows from (<ref>) by observing that a_i + a_j = 1 for all i,j when a_i = 1/2 ∀ i. 
 
For the work that follows, 𝒜=(1/2,1/2, …, 1/2) for M_0,𝒜, which we denote by M_0,( 1/2)^n.



§ CLOSED FORMULA FOR INTERSECTIONS OF Ψ-CLASSES ON M_0,( 1/2)^N

In this section, we develop the closed formula for integrals of ψ̂ monomials corresponding to the ψ̅ monomials on M_0,( 1/2)^n.
 
We denote by = {P_1,P_2,..,P_j,S_1,S_2,...,S_i   } an unordered partition of [n], such that cardinality of each subset in the partition is either 2 or 1; if the cardinality of such a subset is 2, we denote it with P_j and if the cardinality of such subset is 1, we denote it with S_j . The elements of P_j are denoted by P_j1 and P_j2. Further, we denote by F the set of all P_j's, and by S the set of all S_j's. The set of all such partitions is denoted by 𝔓.  


Given a ∈𝔓, we define the graph Γ_ as follows:


  * Γ_ has a `central' node, with |F| number of edges with nodes on ends

  * Attach to each non-`central' node two half edges forming a `fork'; to the `central' node, attach |S| number of half-edges

  * Label the half-edges on a fork as P_j1 and P_j2, and half-edges on central node as S_j's.

So, each P_j corresponds to a fork and S_j's correspond to half-edges on the central node. We call this a -graph. |F| gives the number of forks on the graph. Each such graph is a dual graph of a stratum Δ_ in M_0,n. 

Clearly, the set of all -graphs as defined above are in bijection with the set of all partitions ∈𝔓.

 
Given a ψ̂-monomial m = ψ̂_1^k_1ψ̂_2^k_2ψ̂_3^k_3...ψ̂_r^k_r, a decorated -graph Γ_^dis obtained by coloring a half-edge corresponding to point t ∈ P_j or t ∈ S_j if k_t ≠ 0. 


Given m = ψ̂_1^2ψ̂_2 on M_0,6, 

for = {1,2},{3},{4},{5},{6}, we get the decorated graph Γ_^d as in figure <ref>;
 for = {1,3},{2,4},{5},{6}, we get the decorated graph Γ_^d as in figure <ref>,
 and for = {1,3},{5,6},{2},{4}, we get the decorated graph Γ_^d as in figure <ref>.



 
Consider a stratum Γ_, corresponding to the image of the following gluing morphism:

    gl: M_0,P_1∪⋆ P_1 ×…×M_0,P_s∪⋆ P_s ×M_0,S∪∙ P_1 ∪…∪∙ P_s →M_0, n ,

where for each P_j, the half-edge ∙ P_j is the mark on the node pulled back from the factor corresponding to the central node, and ⋆ P_j is the mark on the node pulled back from the factor corresponding to the respective fork corresponding to a P_j. 

Consider the projection f to the last factor in the product  

    f: M_0,P_1∪⋆ P_1 ×…×M_0,P_s∪⋆ P_s ×M_0,S∪∙ P_1∪…∪∙ P_s →M_0,S∪∙ P_1∪…∪∙ P_s .

We define:

    ∏_j=1^sψ_∙ P_j^k_j[Γ_] :=gl_∗ f^∗( ∏_j=1^sψ_∙ P_j^k_j)[Γ_].




The notation introduced  in  Definition <ref> is an abuse which is convenient enough to be commonly accepted in the field. The class defined in (<ref>) is not a product of the class of the stratum [Γ_] with another class in M_0,n; it is a class supported on the cycle Γ_, and at a basic level one may think of  the symbol ∏_j=1^sψ_∙ P_j^k_j[Γ_] just as a reminder of this fact. However, the notation becomes useful when performing intersection theoretic computations on such classes. Consider two such classes  ∏_j=1^sψ_∙ P_j^k_j[Γ_] and ∏_j=1^s̃ψ_∙P̃_j^k̃_j[Γ_]. One may show that their product is  ∏_j=1^sψ_∙ P_j^k_j∏_j=1^s̃ψ_∙P̃_j^k̃_j[Γ_∩Γ_] if the two strata intersect transversely, and it is otherwise corrected by the euler class of the obstruction bundle that computes the product [Γ_][Γ_] as a cycle pushed forward from the inclusion of Γ_∩Γ_ into the ambient space. For this reason the expression (∏_j=1^sψ_∙ P_j^k_j[Γ_]) (∏_j=1^s̃ψ_∙P̃_j^k̃_j[Γ_])
is often formally manipulated as a product, and rewritten as
 ∏_j=1^sψ_∙ P_j^k_j∏_j=1^s̃ψ_∙P̃_j^k̃_j[Γ_][Γ_].


 
Consider a  -graph as illustrated in Figure (<ref>). For this -graph, define the following ψ-function :

    ϕ_(ψ)=   ψ_S_1^k_S_1… ... ψ_S_q^k_S_q(ψ_∙ P_1^k_P_11+k_P_12-1…ψ_∙P_l^k_P_l1+k_P_l2-1)  .






On M_0,n, let D_i be a divisor D({i,j}) for some j ≠ i. Then any non-trivial ∏_i=1^s D_i, with s ≤ n-3, is supported on a -graph Γ_. 



 We prove by induction.
D_1 = 
[ < g r a p h i c s > ] is clearly a -graph, hence trivially supported on a -graph. Now,

    D_1D_2 = [ < g r a p h i c s > ][ < g r a p h i c s > ]


    = 
      
        0, 
                               if |{1,i_1}∩{2,i_2}|=1 
    [ < g r a p h i c s > ],       
                               if |{1,i_1}∩{2,i_2}|=∅
    
        - [ < g r a p h i c s > ], 
                               if {1,i_1}={2,i_2}

The first is a trivial intersection, the second is a transverse intersection, and the third is a non-transverse intersection. In the case of non-transverse intersection, we decorate the graph with a ψ-class which here we denote by ψ_∙ 1, which corresponds to ψ_∙ P_1 as in Definition <ref>. 
So, all the non-trivial intersections D_1D_2 are supported on -graphs. Now suppose ∏_i=1^s D_i is supported on a -graph, and suppose that this -graph has j number of forks. Then,  

    (∏_i=1^s D_i)D_s+1
 is supported on 

    [ < g r a p h i c s > ][ < g r a p h i c s > ]


    =
      
                    0,                 if{s+1,k}⊈ P_i, S ∀ i 
    [ < g r a p h i c s > ], 
                                                       if{s+1,k}⊆ S 
    
                    - [ < g r a p h i c s > ],
                                       if{s+1,k}⊆ P_i for some i

Again here, the first is a trivial intersection, the second is a transverse intersection, and the third is a non-transverse intersection; and in the case of non-transverse intersection, we decorate the graph with a ψ-class which here we denote by ψ_∙ i, corresponding to subset P_i in the partition of [n] which corresponds to ψ_∙ P_i as in Definition <ref>.
So, all graphs that we get for non-trivial intersections are in fact -graphs. 






The following definition is a special case of Definition <ref>, that will be of particular use in Lemma <ref>.

Given a divisor D({i,j}) on M_0,n, consider the following commutative diagram:

    0.5in0.6inM_0,n
    D({i,j}) [u]^gl[r]^f    M_0, [n]∖{i,j}⋃∙{i,j}

where gl is a gluing morphism, and f is a projection map to M_0,n-1 with marked points [n]∖{i,j}⋃∙{i,j}.
Then,

    ψ_∙{ i,j}^k D({i,j}) := gl_∗ f^∗ψ_∙{ i,j}^k.






We use notation as in Definitions <ref>, <ref> and intersection theoretic calculus as explained in Remark <ref>; on M_0,n, we have

    ψ̂_i^k_i =  ψ_i^k_i- ∑_jψ_∙{ i,j}^k_i-1 D({i,j})



Using Corollary <ref>,

    ψ̂_i^k_i =  (ψ_i- ∑_j D({i,j}))^k_i


    =   ψ_i^k_i + … + (-1)^k_i( ∑_j D({i,j})) ^k_i


    =   ψ_i^k_i + (-1)^k_i∑_j D({i,j})^k_i


    =  ψ_i^k_i + … + (-1)^k_i (-1)^k_i-1ψ_∙{ i,j}^k_i-1∑_j D({i,j})


    =  ψ_i^k_i- ∑_jψ_∙{ i,j}^k_i-1 D({i,j})

The third equality happens as all the terms in the expansion except the first and the last vanish due to dimension reasons. Furthermore, it's only the self intersections that are non-zero in the last term in the expansion of (ψ_i- ∑_j D({i,j}))^k_i. The fourth and fifth equalities follow from the k_i self intersections of ∑_j D({i,j}). 


For the result in the above lemma, we will use the following notation for brevity. 

    ψ̂_i^k_i =  ψ_i^k_i- ψ_∙_i^k_i-1 D_i

where 

    D_i := ∑_jD({i,j})

and 

    ψ_∙_i^k_i-1 D_i := ∑_jψ_∙{ i,j}^k_i-1 D({i,j})

Also, again for brevity, if the -graph in figure (<ref>) corresponds to partition = {P_1,P_2,..,P_s,S_1,S_2,...,S_q   }, then we will denote ψ_∙ P_j also as ψ_P_j in what follows.


With , F, S, P_j,P_j1, P_j2, S_i as defined above, for n≥ 5 we have for M_0,n:

    ψ̂_1^k_1ψ̂_2^k_2…ψ̂_n^k_n =
    ∑_∈𝔓 (-1)^|F|[  Γ_] ∏_S_i ∈ Sψ_S_i^k_S_i∏_P_j ∈ Fψ_P_j^k_P_j1 + k_P_j2 - 1

where [  Γ_] is the class of boundary stratum in M_0,n with dual graph Γ_.


Let D_i = ∑_jD({i,j}). Omitting ψ̂'s with 0-exponents, and assuming, without loss of generality, that the first r number of ψ̂'s remain with nonzero-exponents,  
ψ̂_1^k_1ψ̂_2^k_2…ψ̂_n^k_n 
= ψ̂_1^k_1ψ̂_2^k_2ψ̂_3^k_3...ψ̂_r^k_r
. Then, 

    ψ̂_1^k_1ψ̂_2^k_2ψ̂_3^k_3...ψ̂_r^k_r =  (ψ_1- D_1)^k_1(ψ_1- D_2)^k_2… (ψ_r-  D_r)^k_r
 using relation (<ref>)

    = (ψ_1^k_1- ψ_∙_1^k_1-1 D_1)
            (ψ_2^k_2-  ψ_∙_2^k_2-1 D_2) …
            (ψ_r^k_r-  ψ_∙_r^k_r-1 D_r)

        using relation (<ref>) 

    =(-1)^s ∑_s=0^rψ_∙_i_1^k_i_1-1ψ_∙_i_2^k_i_2-1…ψ_∙_i_s^k_i_s-1 D_i_1 D_i_2… D_i_sψ_i_s+1^k_i_s+1…ψ_i_r^k_i_r

with 1 ≤ i_j ≤ r and i_1 < i_2 < … < i_s. Now, each term in the expansion of the expression above is supported on a -graph from Lemma <ref>. 
Pick a -graph with s number of colored half-edges on forks and form the corresponding Γ_^d. There are two possibilities: 1) at least one fork has both half-edges uncolored, or 2) all forks of Γ_^d have at least one colored half-edge. Let the second type of graph (figure <ref>) have t number of forks with both half-edges colored as shown below. Here, ={  P_1, P_2, …, P_s-t/2, S_1,…, S_r-s} = {{i_1, i_2}, …, {i_s,j_s-t}, i_s+1,…, i_r}. 

Intersect both types of -graph with ϕ_(ψ) as defined in (<ref>), which for this -graph is :


    ϕ_(ψ)= (ψ_∙ 1 ^k_i_1+k_i_2-1…ψ_∙t/2^k_i_t-1+k_i_t-1ψ_∙_t/2+1^k_i_t+1-1…ψ_∙s-t/2^k_i_s-1) ψ_i_s+1^k_i_s+1… ... ψ_i_r^k_i_r

Claim : The intersection of ϕ_(ψ) with the first type of graphs gives 0.

Proof Without loss of generality, suppose the first type of -graph have i_1 and i_2 on a fork with k_i_1=k_i_2 = 0. Then k_i_1+k_i_2-1= -1 and 

    ϕ_(ψ)= ψ_∙_1^-1…ψ_∙t/2^k_i_t-1+k_i_t-1ψ_∙_t/2+1^k_i_t+1-1…ψ_∙s-t/2^k_i_s-1ψ_i_s+1^k_i_s+1… ... ψ_i_r^k_i_r

is 0 as negative power of a ψ class by standard convention is 0. 
Now, consider the second type of  -graph (Figure <ref>).

Claim: ϕ_( ψ) ·Γ_^d, where Γ_^d is the second type of graph, uniquely determines a term in the expansion (<ref>) above.
 
Proof: 
Define a map from the set of -graphs to the terms in the expansion (<ref>) as follows. For each i_l in Γ_^d where i_l is a colored half-edge on a fork, assign a D_i_l, to form the product ∏_l=1^s D_i_l. Now, for each fork on the graph with half-edges i_l and i_q, assign ψ_∙_m^k_i_l+k_i_q-1 and form their product; the result is ϕ_(ψ). Then ϕ(ψ)·∏_l=1^s D_i_l is precisely the term in the expansion (<ref>) that Γ_^d maps to. Further, reversing the process, we get the preimage of a term in expansion (<ref>) the unique -graph (figure <ref>). So, the map is in fact a bijection. Now, 


    ϕ_( ψ)·[Γ_^d] =


    (-1)^s+t/2[ Γ_] (ψ_∙ 1 ^k_i_1+k_i_2-1…ψ_∙t/2^k_i_t-1+k_i_t-1ψ_∙_t/2+1^k_i_t+1-1…ψ_∙s-t/2^k_i_s-1) ψ_i_s+1^k_i_s+1… ... ψ_i_r^k_i_r


    = (-1)^|F|[  Γ_]  ∏_S_i ∈ Sψ_S_i^k_S_i∏_P_j ∈ Fψ_P_j^k_P_j1 + k_P_j2 - 1 .





§ NUMERICAL INTERSECTIONS

In this section, we develop two corollaries of theorem (<ref>) for two versions of a closed formula for top intersections of ψ̂-classes on M_0,n.Then we encode this formula in a generating function obtained by applying a differential operator to the Witten potential. As pointed out earlier, these corollaries (<ref> and <ref>) can also be deduced from theorem 7.9 in <cit.>. For our work, we develop specific and explicit closed formulas here and base our combinatorial analysis closely on the structure of dual graphs.   



With , F, P_j, P_j1, P_j2, S_i as defined above, for n≥ 5 we have:

    ∫_M_0,nψ̂_1^k_1ψ̂_2^k_2…ψ̂_n^k_n =
    ∑_∈𝔓 (-1)^|F|||-3⟨k_P_i1+k_P_i2-1⟩,⟨ k_S⟩

where ∑ k_i = n-3, and 
for a ={  P_1, P_2, …, P_s, S_1,…, S_q}, 
⟨k_P_i1+k_P_i2-1⟩= k_P_11+k_P_12-1, …,k_P_s1+k_P_s2-1, and ⟨ k_S⟩ = k_S_1,…,k_S_q.


The proof of this is same as for theorem (<ref>) except in the last part of evaluation of ϕ_( ψ)·[  Γ_]. Here when ∑ k_i = n-3, this evaluation gives
  

    ϕ_( ψ)·[  Γ_]


    =(-1)^s+t/2n-3-(s-t/2)k_i_1+k_i_2-1, …,k_i_t+1-1,…,k_i_s-1,k_i_s+1,…, k_i_r


    =(-1)^|F|n-3-|F|k_i_1+k_i_2-1, …,k_i_s+k_j_s-1,k_i_s+1,…, k_i_r

as k_i_r+1=…= k_i_n=0

    = (-1)^|F|||-3⟨k_P_i1+k_P_i2-1⟩,⟨ k_S⟩.


We can reduce the complexity of the computation of (<ref>) if we can remove from 𝔓 those partitions 𝒫's whose graphs evaluate to 0 when intersected with ϕ(ψ).
Also, we can collect together terms that come from permuting the marked points j_1,j_2,...,j_s-t in the partition 𝔓 as all these terms evaluate to the same value as k_j_i=0 for all these j_i.

Form a new set 𝔓' in the following way: 
Make the powerset ℜ of [r], where r denotes the number of ψ's with non-zero exponent in the ψ-monomial. For each set ∈ℜ, form all subsets 𝒫' of  whose elements have cardinality 2 or 1 with the upper bound of number of subsets of cardinality 2 fixed at n/2. Call 𝔓' the set of all 𝒫'.
This set 𝔓' can also be obtained from 𝔓 via the following map: 
Given a partition 𝒫, project to a 𝒫' by forgetting all points S_i ∈𝒫 and in a P_j ∈𝒫 forget a point P_ji if k_P_ji=0. More formally, 𝒫'= {P_1',P_2',..,P_i',S_1',S_2',..S_j'} where P_i' = P_i if k_Pi1>0 and k_Pi2>0; S_i' = P_i ∖{P_ij} if k_P_ij=0. This is an onto map. Each P_j^' has cardinality 2, and each S_i' has cardinality 1. Denote by 𝒫'^c the set [r]∖ P'_1 ∪ ...∪ P'_i∪ S_1'∪ S_2' ∪..∪ S_j'.

With 𝔓' as defined above, for n≥ 5 we have:

    ∫_M_0,n  ψ̂_1^k_1ψ̂_2^k_2ψ̂_3^k_3...ψ̂_r^k_r =
    ∑_𝒫' ∈𝔓' (-1)^s+t/2(n-r)!/(n-r-s+t)!n-3-(s-t/2)k_i_1+k_i_2-1,..,k_i_t+1-1,..,k_i_s-1,k_c_1,.., k_c_q

where t= 2|{ P_i^'}|, s= |{ S_i'}| + t and {c_1,..,c_q}=𝒫'^c
 


Corresponding to a partition 𝒫, form a corresponding decorated graph for 𝒫^' by uncoloring any half-edges on the central node, and `forgetting' the j_i's on uncolored half-edges on the forks as discussed above. This corresponds to 𝒫' = { P_1^', P_2^', …, P_t/2^', S_1^',…, S_s-t^'}, where P_i^''s correspond to nodes with both half-edges colored, and S'_j = i_t+j in the decorated graph of 𝒫 in figure (<ref>). Now consider intersection of this graph with ϕ_(ψ) as defined earlier in (<ref>) :
 
ϕ_(ψ) ·[ < g r a p h i c s > ]

    ≅M_0,3×M_0,3,…,M_0,3×ϕ(ψ). M_0,n-3-(s-t/2)


    =(-1)^s+t/2n-3-(s-t/2)k_i_1+k_i_2-1, ..,k_i_t-1+k_i_t-1,k_i_t+1-1,..,k_i_s-1,k_c_1,.., k_c_q

As there are (n-r)!/(n-r-s+t)! ways of choosing the uncolored half-edges on the forks, corresponding to j_i's, the term evaluates to 

    =(-1)^s+t/2(n-r)!/(n-r-s+t)!n-3-(s-t/2)k_i_1+k_i_2-1, ..,k_i_t+1-1,..,k_i_s-1,k_c_1,.., k_c_q




 §.§ Generating Function for the top intersections

We start with the generating function- Witten potential(<cit.>)- also called, in topological gravity, the total free energy. The correlation functions are defined as intersection numbers on the moduli space of stable n-pointed curves (here for genus 0) as
 

    ⟨τ_k_1…τ_k_n⟩ :=∫_M_0,nψ_1^k_1ψ_2^k_2…ψ_n^k_n
   
Collecting all tau's with equal exponent, we can write 
⟨τ_k_1…τ_k_n⟩ = 
⟨τ_0^s_0τ_1^s_1τ_2^s_2…τ_m^s_m⟩. Now, define s = (  s_0, s_1, …), 
 and 
⟨τ^𝐬⟩ := ⟨τ_0^s_0τ_1^s_1τ_2^s_2…τ_m^s_m⟩. So, for each sequence s, there is a correlation function ⟨τ^𝐬⟩; and |𝐬| := ∑s_i is the number of marks n. For the generating function, all these correlation functions are collected and used as coefficients in a formal power series. Using notation 𝐭^𝐬 = ∏_i=0^∞ t_i^s_i, and 𝐬! = ∏_i=0^∞s_i!, the generating function is

    F(𝐭) := ∑_𝐬𝐭^𝐬/𝐬!⟨τ^𝐬⟩


    = (1) t_0^3t_1/3! 
    + (1) t_0^4t_2/4! + (2) t_0^3t_1^2/3!2!
    + (1) t_0^5t_3/5!+ (3) t_0^4t_1t_2/4!+ (6) t_0^3t_1^3/3!3! +……


where the coefficients of appropriate terms give the intersection numbers ∫_M_0,nψ_1^k_1ψ_2^k_2…ψ_n^k_n. Observe that  the total codimension of the integrand in ⟨τ^𝐬⟩ is ∑ is_i, so |s| - 3 = ∑ i s_i. With this generating function, the String equation (<ref>) for M_0,n is encoded in the differential equation -


    ∂/∂ t_0 F = t_0^2/2 +  ∑_i=0^∞ t_i+1∂/∂ t_i F




Define a new generating function 

    G(𝐭) := ∑_𝐬𝐭^𝐬/𝐬!⟨τ̂^𝐬⟩

where ⟨τ̂^𝐬⟩ = ∫_M_0,nψ̂_1^k_1ψ̂_2^k_2…ψ̂_n^k_n, and ∑ k_i = n-3. 

So, G(𝐭) has as coefficients of the monomials 𝐭^𝐬/𝐬! the intersection numbers ∫_M_0,nψ̂_1^k_1ψ̂_2^k_2…ψ̂_n^k_n for any value of n and any values of k_i's, with ∑ k_i = n-3.

Given two operators 𝒥_1 = t_pt_q∂/∂ t_r, and 𝒥_2 = t_p't_q'∂/∂ t_r', 
the normal ordering of the product, denoted : 𝒥_1 𝒥_2 :, means that we treat the t_i's and ∂/∂ t_i's as commuting variables, and bring all t_i's to the left of ∂/∂ t_i's. E.g., here 

    𝒥_1 𝒥_2  = t_pt_q∂/∂ t_r t_p't_q'∂/∂ t_r',
 but

    : 𝒥_1 𝒥_2 : = t_pt_q t_p't_q'∂/∂ t_r∂/∂ t_r'.

Furthermore, 

    : 𝒥_1  +  𝒥_2 : =  : 𝒥_1 : + : 𝒥_2 :


It follows from the above definition that normal ordering extends to arbitrary products of operators, and extends by linearity to arbitrary sums (including infinite sums) of operators. In particular, it extends to : e^𝒥 :. 
 

With G(t) as defined above, 

    G(t) = (F(t)) - t_0^3/3! + 2t_0^3 t_1/3!

where =  : e^-  :, and : e^-  : denotes the operator with normal ordering, and 

    = 1/2∑_i,j=0^∞ t_it_j∂/∂ t_i+j-1


Before we prove the above theorem, consider the ψ̂-monomial ∫_M_0,nψ̂_1^n-3. It is the coefficient of t-monomial t_0^(n-1)t_n-3/(n-1)! in G(t). When we apply the operator = 1 -  + : ^2/2!: - … to F(t), only the  terms with the following t-monomials in F(t) contribute to the term with t-monomial t_0^(n-1)t_n-3/(n-1)! in (F(t)):
t_0^(n-1)t_n-3/(n-1)! and t_0^(n-2)t_n-4/(n-2)!.
The corresponding operations happen as follows. 
The first term of , which is 1, acts on F(t) to produce the t-monomial t_0^(n-1)t_n-3/(n-1)! as it is. Then, among the summands in -, the only operator that produces the t-monomial t_0^(n-1)t_n-3/(n-1)! is -t_0t_n-3∂/∂ t_n-4, which acts as follows. 

    -t_0t_n-3∂/∂ t_n-4(  t_0^(n-2)t_n-4/(n-2)!)  = -(n-1) t_0^(n-1)t_n-3/(n-1)! 
    .
 

No other term in F(t) contributes to the coefficient of the t-monomial t_0^(n-1)t_n-3/(n-1)! in (F(t)). As the monomial t_0^(n-1)t_n-3/(n-1)! has the coefficient ∫_M_0,nψ_1^n-3, and the monomial t_0^(n-2)t_n-4/(n-2)! has the coefficient ∫_M_0,n-1ψ_1^n-4 in F(t), the coefficient of t_0^(n-1)t_n-3/(n-1)! in  (F(t)) is 
 ∫_M_0,nψ_1^n-3 - (n-1) ∫_M_0,n-1ψ_1^n-4, which equals ∫_M_0,nψ̂_1^n-3 from Corollary (<ref>).
Observe that both the contributions correspond to the two types of 𝒫-graphs that make non-zero contributions to ∫_M_0,nψ̂_1^n-3 in corollary (<ref>). The first are of the type with no forks, and the second are of the type with one fork. 

For each 𝒫-graph, the corresponding 𝒫_k-graph is defined by replacing each i ∈ [n] on the 𝒫-graph by k_i.  

Clearly the map {𝒫-graphs}→{𝒫_k-graphs} is a surjection.


In genus 0, for a given ⟨τ_k_1…τ_k_n⟩ =∫_M_0,nψ_1^k_1ψ_2^k_2…ψ_n^k_n as defined above, let 
s = (  s_0, s_1, … s_n) be such that ⟨τ_k_1…τ_k_n⟩ = 
⟨τ_0^s_0τ_1^s_1τ_2^s_2…τ_n^s_n⟩. 
Consider a 𝒫_k-graph with m forks with q distinct k_i's appearing on the forks; let such k_i's be {  k_1, k_2, … k_q}. Let l_i be the number of times a given k_i appears on any fork on the 𝒫_k-graph. Then the number of 𝒫-graphs that map to this 𝒫_k-graph is given by:

    1/|Aut(𝒫̂_k)|s_k_1!/( s_k_1-l_1)! s_k_2!/( s_k_2-l_2)! …s_k_q!/( s_k_q-l_q)!

where |Aut(𝒫̂_k)| is the number of automorphisms of the subgraph of the 𝒫_k-graph obtained by removing half-edges on the central node.


Consider a 𝒫_k-graph with m forks such that the number of k_i's appearing on the forks is q. Then, if all n half-edges are given ordering, the number of corresponding -graphs would be 

    s_0! s_1! s_2! … s_n!.

Now, we divide by the permutations of half-edges on the central node to get 

    s_0!/( s_0-l_0)! s_1!/( s_1-l_1)! s_2!/( s_2-l_2)! …s_n!/( s_n-l_n)! .

Now, as only q number of k_i's appear on the forks, l_i = 0 for i > q, so 

    s_0!/( s_0-l_0)! s_1!/( s_1-l_1)! …s_n!/( s_n-l_n)!  = s_k_1!/( s_k_1-l_1)! s_k_2!/( s_k_2-l_2)! …s_k_q!/( s_k_q-l_q)! .

Further we need to divide by permutations of half-edges on the forks. Let j_1, j_2, …, j_f be the number of forks with the same set of k_i's on them; and let d be the number of forks with both k_i's same on that fork. Then, we divide by 2^d (  j_1! j_2! … j_f!) to get 

    ( 1/2^d (  j_1! j_2! … j_f!)) s_k_1!/( s_k_1-l_1)! s_k_2!/( s_k_2-l_2)! …s_k_q!/( s_k_q-l_q)! .
 
Observe that the number (2^d (  j_1! j_2! … j_f!) ) is the number of automorphisms of the subgraph of 𝒫_k-graph obtained by removing the half-edges on the central node; denote this subgraph as 𝒫̂_k. Then the number of 𝒫-graphs that map to this 𝒫_k-graph can be rewritten as:

    1/|Aut(𝒫̂_k)|s_k_1!/( s_k_1-l_1)! s_k_2!/( s_k_2-l_2)! …s_k_q!/( s_k_q-l_q)!
 
which we denote by C__k.

The reason for organizing C__k as in (<ref>) will become clear in the proof of theorem (<ref>).


With the definitions and notations above, Corollary (<ref>) can be rewritten as :

    ∫_M_0,nψ̂_1^k_1ψ̂_2^k_2…ψ̂_n^k_n =
    ∑__k∈𝔔 (-1)^m C__k∫_M_0,(n-m)ψ_∙_1^k_i_1+k_i_2-1…ψ_∙_m^k_i_2m-1+k_i_2m-1ψ_i_2m+1^k_i_2m+1… ... ψ_i_r^k_i_r

where m is number of forks on the 𝒫_k-graph and C__k is the number of 𝒫-graphs that map to this 𝒫_k-graph, and 𝔔 is the set of all 𝒫_k-graphs.
 

This version of closed formula for ∫_M_0,nψ̂_1^k_1ψ̂_2^k_2…ψ̂_n^k_n is just a reorganization of (<ref>) using 𝒫_k-graphs instead of 𝒫-graphs. As the map {𝒫-graphs}→{𝒫_k-graphs} is a surjection, we get all the terms in (<ref>).

Now, for a general 𝒫_k-graph with m forks shown below, we define the following operator (which appears in ):

    𝒟__k := t_k_i_1 t_k_i_2… t_k_i_2m-1 t_k_i_2m∂/∂ t_k_i_1+k_i_2-1…∂/∂ t_k_i_2m-1+k_i_2m-1







By construction, the operators (<ref>) are in bijection with the 𝒫_k-graphs. Furthermore, the operator (<ref>) arises in  as a summand in :(-1)^m^m/m!: with some multiplicity. As part of the proof of theorem, we will show that this multiplicity is (-1)^m 1/|Aut(𝒫̂_k)| with |Aut(𝒫̂_k)| as defined in Lemma (<ref>). 
















Strategy of proof of theorem (<ref>): we will show that for a general t-monomial,
its coefficients in G(t) and (F(t)) - t_0^3/3! + 2t_0^3 t_1/3! are equal. For this, we fix a t-monomial, and show a bijection and equality between the summands in the right hand side of (<ref>) and the summands in coefficient of the t-monomial in (F(t)) - t_0^3/3! + 2t_0^3 t_1/3!. This is done via the _k graphs as follows. Each _k graph tautologically gives a summand in (<ref>). Each _k graph also gives a term in  (with some multiplicity) and a monomial in F(t) whose product is the same t-monomial, and its coefficient
in this product agrees with the summand in (<ref>). Finally, all the terms in
(F(t)) - t_0^3/3! + 2t_0^3 t_1/3!  are of the form obtained above.

Before the proof, here is an example that illustrates the idea.


Consider
 

    ∫_M_0,61ψ̂_1^1…ψ̂_8^1ψ̂_9^2…ψ̂_13^2ψ̂_14^3ψ̂_15^3ψ̂_16^4ψ̂_17^4ψ̂_18^4ψ̂_19^5ψ̂_20^5ψ̂_21^6ψ̂_22^6.
 
The corresponding  t-monomial in G(t) is 

    t_0^s_0t_1^s_1…t_l^s_l/s_0!s_1! …s_l! = t_0^39t_1^8t_2^5t_3^2t_4^3t_5^2t_6^2/39!8! 5! 2! 3! 2! 2! =: T.

Now consider the following 𝒫_k-graph in figure (<ref>) with m=7 forks.


The corresponding operator (<ref>) is

    𝒟__k = t_1^7 t_2^5 t_3 t_4 ∂^3/∂ t_2^3∂^2/∂ t_1^2∂/∂ t_3∂/∂ t_6.
 
The coefficient of this operator in : (-1)^7 ^7/7! : is 

    (-1)^7 (  1/2^7 7!)  ( 2^473,2,1,1)
    
     = (-1)^7 (  1/2^3 3! 2!)

The corresponding unique t-monomial T̃__k in F(t) such that 𝒟__k (T̃__k) = T is 

    t_0^39t_1^3t_2^3t_3^2t_4^1t_5^2t_6^3/39!3! 3! 2! 1! 2! 3! =: T̃__k

Observe that T̃__k can also be read off from the 𝒫_k-graph. 
Now, in  (F(t)), the corresponding term is 

    (-1)^7 (  1/2^3 3! 2!) ( ⟨τ^𝐬⟩__k)  𝒟__k (T̃__k)

where  

    ⟨τ^𝐬⟩__k = ∫_M_0,54ψ̂_1^1ψ̂_2^1ψ̂_3^1ψ̂_4^2ψ̂_6^2ψ̂_7^3ψ̂_8^3ψ̂_9^4ψ̂_10^5ψ̂_12^5ψ̂_13^6ψ̂_14^6ψ̂_15^6.

Again, ⟨τ^𝐬⟩__k can also be read off from the 𝒫_k-graph. Also, observe that the coefficient (-1)^7 (  1/2^3 3! 2!) of 𝒟__k in :(-1)^7^7/7! : is exactly  (-1)^m 1/|Aut(𝒫̂_k)| as claimed earlier. Now, 

    (-1)^7 (  1/2^3 3! 2!) ( ⟨τ^𝐬⟩__k)  𝒟__k (T̃__k)


    = (-1)^7 ( 1/2^3 3! 2!) ( 8!/(8-7)!5!/(5-5)!2!/(2-1)!2!/(2-1)!3!/(3-1)!)  ( ⟨τ^𝐬⟩__k) T


    = (-1)^7 ( 1/2^3 3! 2!)  ( 8! 5! 2! 2!3! /2!) ( ⟨τ^𝐬⟩__k) T


    =(-1)^7 ( 1/|Aut(𝒫̂_k)|)  ( 8! 5! 2! 2!3! /2!) ( ⟨τ^𝐬⟩__k) T


    = - C__k( ⟨τ^𝐬⟩__k) T

which gives a summand in the coefficient of T in (F(t))- t_0^3/3! + 2t_0^3 t_1/3!, and this summand agrees with  the summand in (<ref>) corresponding to the chosen _k-graph.










 (of theorem (<ref>))


















Consider a t-monomial 
t_0^s_0t_1^s_1…t_l^s_l/s_0!s_1! …s_l! =: T. In
G(t), T has the coefficient ∫_M_0,nψ̂_1^k_1ψ̂_2^k_2…ψ̂_n^k_n. Now, we will show bijection and equality between the summands in the right hand side of (<ref>) and the summands in coefficient of the T in (F(t))- t_0^3/3! + 2t_0^3 t_1/3!.


  * Pick a 𝒫_k-graph with no fork. The corresponding operator (<ref>) in  is 1, the first term in , which when applied to F(t) results in coefficient 1 for t_0^s_0t_1^s_1…t_l^s_l/s_0!s_1! …s_l! in  (F(t)), which agrees with the corresponding summand in (<ref>).

  * Pick a 𝒫_k-graph with one fork. Without loss of generality, assume a 𝒫_k-graph with k_1, k_2 on the only fork, with k_1, k_2 not simultaneously 0.

 
Case 1 : k_1 ≠ k_2.
Then, the corresponding operator (<ref>) is t_k_1 t_k_2∂/∂ t_k_1+k_2-1. In , this term has coefficient -1. The term in F(t) that it operates on to produce  t_0^s_0t_1^s_1…t_l^s_l/s_0!s_1! …s_l! has t-monomial: 


    t_0^s_0…t_k_1^s_k_1-1…t_k_2^s_k_2-1…t_k_l^s_k_l/s_0! …s_k_1-1! …s_k_2-1! …s_l! t_k_1+k_2-1.
 

The result of applying -t_k_1 t_k_2∂/∂ t_k_1 + k_2-1 in  to F(t) is the following:  

    -t_k_1 t_k_2∂/∂ t_k_1 + k_2-1 (t_0^s_0…t_k_1^s_k_1-1…t_k_2^s_k_2-1…t_k_l^s_k_l/s_0! …s_k_1-1! …s_k_2-1! …s_l! t_k_1+k_2-1)


    = - s_k_1 s_k_2t_0^s_0t_1^s_1…t_l^s_l/s_0!s_1! …s_l!.
  
So, the summand contributed by this operator to the coefficient of t_0^s_0t_1^s_1…t_l^s_l/s_0!s_1! …s_l! in (F(t)) is  

    - s_k_1 s_k_2∫_M_0,(n-1)ψ_∙_1^k_1+k_2-1ψ_3^k_3… ... ψ_r^k_r

and C__k = s_k_1 s_k_2 is the number of 𝒫-graphs that map to this 𝒫_k-graph. This summand also agrees with the corresponding summand in (<ref>).
Case 2 : k_1 = k_2. In this case we get the corresponding operator (<ref>) as 
t_k_1^2  ∂/∂ t_2k_1-1. The coefficient of this term in  is -1/2.
When -1/2 t_k_1^2 ∂/∂ t_2k_1-1 is applied to F(t), the only terms that produces t_0^s_0t_1^s_1…t_l^s_l/s_0!s_1! …s_l! is  

    t_0^s_0t_1^s_1…t_k_1^s_k_1-2…t_k_l^s_k_l/s_0! …s_k_1-2! …s_l! t_2k_1-1

Now,

    - 1/2 t_k_1^2 ∂/∂ t_2k_1-1 (t_0^s_0t_1^s_1…t_l^s_l/s_0!s_1! …s_l!t_0^s_0t_1^s_1…t_k_1^s_k_1-2…t_k_l^s_k_l/s_0! …s_k_1-2! …s_l! t_2k_1-1)

    = - 1/2 s_k_1 (s_k_1-1) t_0^s_0t_1^s_1…t_l^s_l/s_0!s_1! …s_l!

So, the the summand contributed by this operator to the coefficient of t_0^s_0t_1^s_1…t_l^s_l/s_0!s_1! …s_l! in (F(t)) is   

    - 1/2 s_k_1 (s_k_1-1) ∫_M_0,(n-1)ψ_∙_1^2k_1-1ψ_3^k_3… ... ψ_r^k_r

and C__k = 1/2 s_k_1 (s_k_1-1) is the number of 𝒫-graphs that map to this kind of 𝒫_k-graph. This summand also agrees with the corresponding summand in (<ref>). 

In both cases, the terms contribute - C__k∫_M_0,(n-1)ψ_∙_1^k_1+k_2-1ψ_3^k_3… ... ψ_r^k_r to the coefficient of t-term t_0^s_0t_1^s_1…t_l^s_l/s_0!s_1! …s_l! in (F(t)). So, we get both summands in (<ref>) corresponding to two _k-graphs with one fork. Also, observe that if k_1 = k_2 = 0, the term - t_0^2 ∂/∂ t_-1 contributes nothing, and there is also no corresponding summand in (<ref>).

Now consider a 𝒫_k-graph with m forks. Without loss of generality, let the k_i's on the forks be { k_1, k_2, …, k_2m} as shown in figure below. Let l_i be the number of times a given k_i appears on any fork on the 𝒫_k-graph, and let j_1, j_2, …, j_f be the number of forks with the same set of k_i's on them; and let d be the number of forks with both k_i's the same on that fork.



 




Then the corresponding operator (<ref>) is

    t_k_1 t_k_2…  t_k_2m-1 t_k_2m∂/∂ t_k_1+k_2-1…∂/∂ t_k_2m-1+t_k_2m-1 := 𝒟_𝒫_k.


The coefficient of this term in  as a summand in  
 :(-1)^m^m/m!: is given by

    (-1)^m (  1/2^m m!)  ( 2^(m-d)ml_1,l_2,…,l_n)
    
     = (-1)^m (  1/2^d l_1! l_2! …,l_n!).

The corresponding term in F(t) is 

    t_0^s_0…t_k_1^( s_k_1- l_k_1)…t_k_2m^( s_k_2m- l_k_2m)/s_0! …( s_k_1- l_k_1)!  …( s_k_2m- l_k_2m) ! …s_k_n!
    t_k_1+k_2-1… t_k_2m-1+k_2m-1
    =: T̃__k.

In  (F(t)), the corresponding term is 

    (-1)^m (  1/2^d l_1! l_2! …,l_n!)  ( ⟨τ^𝐬⟩__k)  𝒟__k (T̃__k)

where  
⟨τ^𝐬⟩__k is the ψ-monomial on M_0,(n-m) that appears as coefficient of T̃__k in F(t). 
Observe that the coefficient (-1)^m (  1/2^d l_1! l_2! …,l_n!) of 𝒟__k in :(-1)^m^m/m!: is exactly  (-1)^m 1/|Aut(𝒫̂_k)| as claimed earlier. Now, 

    (-1)^m (  1/2^d l_1! l_2! …,l_n!)  ( ⟨τ^𝐬⟩__k)  𝒟__k (T̃__k)


    = (-1)^m (  1/2^d l_1! l_2! …,l_n!)  ( s_1!/( s_1-l_1)! s_2!/( s_2-l_2)! …s_n!/( s_n-l_n)! )  ( ⟨τ^𝐬⟩__k) T


    = (-1)^m ( 1/|Aut(𝒫̂_k)|)  ( s_1!/( s_1-l_1)! s_2!/( s_2-l_2)! …s_n!/( s_n-l_n)! )  ( ⟨τ^𝐬⟩__k) T


    = (-1)^m C__k( ⟨τ^𝐬⟩__k) T

which gives a summand in the coefficient of T in (F(t)) and this agrees with  the summand in (<ref>) corresponding to the chosen _k-graph.


So, one direction is proved. To show bijection in the other direction, we pick a summand in the coefficient of t_0^s_0t_1^s_1…t_l^s_l/s_0!s_1! …s_l! = T in (F(t)) that comes from term :(-1)^m/m!^m :. Let this summand come from the following summand in :(-1)^m/m!^m :

    (-1)^m (  1/2^d l_1! l_2! …,l_n!)  𝒟

where 

    𝒟 = t_k_1 t_k_2…  t_k_2m-1 t_k_2m∂/∂ t_k_1+k_2-1…∂/∂ t_k_2m-1+t_k_2m-1.

Now, 𝒟 = 𝒟__k, with _k as in the figure (<ref>).
Now, ⟨τ^𝐬⟩__k is uniquely determined by T̃__k, and T̃__k is uniquely determined by the relation

    𝒟__k (T̃__k) = T.

Then the chosen summand in the coefficient of T in (F(t)) is 

    = (-1)^m C__k( ⟨τ^𝐬⟩__k).

The _k-graph thus determined also determines the same summand in (<ref>).


Finally, (F(t)) also contains two monomials with degree ≤ 4, and these are  
t_0^3/3! and - 2t_0^3 t_1/3!. We subtract these monomials as the Hassett Spaces for the corresponding cases of three and four pointed rational curves with all weights 1/2 are empty, as these rational curves are not 𝒜-stable.




§ ACKNOWLEDGEMENTS
 
The author would like to thank the anonymous reviewer for constructive criticism and helpful suggestions.


plain

