On the locus of genus $3$ curves that admit meromorphic differentials with a zero of order $6$ and a pole of order $2$
Abel Castorena Quentin Gendron
math.AG

Curves with a differential with singularities of order (6;-2)]On the locus of genus 3 curves that admit
meromorphic differentials with a zero of order 6 and a pole of order 2

Castorena]Abel Castorena



abel@matmor.unam.mx
Research of the first author is supported by Grants PAPIIT UNAM IN100419 "Aspectos Geometricos del moduli de curvas M_g", and CONACyT A1-S-9029 "Moduli de curvas y curvatura en A_g.
Gendron]Quentin Gendron
gendron@matmor.unam.mx
Research of the second author was supported by a Posdoctoral Fellowship from DGAPA, UNAM.

Centro de Ciencias Matemáticas-UNAM, Antigua Car. a Pátzcuaro 8701,
Col. Ex Hacienda San José de la Huerta,
Morelia, Mich., México


 
The main goal of this article is to compute the class of the divisor of [3] obtained by taking the closure of the image of [3](6;-2) by the forgetful map. This is done using Porteous formula and the theory of test curves. For this purpose, we study the locus of meromorphic differentials of the second kind, computing the dimension of the map of these loci to  and solving some enumerative problems involving such differentials in low genus. A key tool of the proof is the compactification of strata recently introduced by Bainbridge-Chen-Gendron-Grushevsky-Möller.





[
    [
    December 30, 2023
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§ INTRODUCTION


Let X be a smooth projective irreducible complex curve of genus g and let K_X be the canonical line bundle on X. The global sections of K_X are the holomorphic differentials, and they form a vector space H^0(X, K_X) of dimension g.  A non-zero holomorphic differential ω over a curve X induces a translation structure on the complement of the zeroes of ω which can be realized as a plane polygon with certain side identifications by translations. Hence the pair (X,ω) is called indistinctly a translation surface or an abelian differential (see for example <cit.>). 
 Let Ωℳ_g be the moduli space of abelian differentials (X,ω) of genus g.
This forms a vector bundle Ωℳ_g →ℳ_g whose fiber 
on X∈ℳ_g is the space H^0(X,K_X), which is  called the Hodge bundle.
Let μ=(a_1,… ,a_n) a positive partition of 2g-2, that is, the integers a_i∈_>0 

satisfy
∑_j=1^n a_j=2g-2. The stratum of abelian differentials (μ) of type μ parametrises all couples (X,ω) ∈
with prescribed zeroes of order a_i at distinct points z_i∈ X for  i=1,… ,n. 
The above construction can be extended to the case of meromorphic abelian differentials. For every partition  μ = (a_1,…,a_n;-b_1,…,-b_p) of 2g-2 with a_i,b_j≥1 there is a moduli space (μ) parametrising the pairs (X,ω), where X is a genus g curve and ω a meromorphic differential with zeroes of order a_i at some points z_i and poles of order b_j at the points w_j.
From the projection map →[g] we have a projective bundle () over [g] with fibre ^g-1. The image of the stratum (μ) in () is called the projective stratum (μ). Similarily we can define  the projective strata in the meromorphic case. In both cases, the  projective strata (μ) parametrize abelian differentials modulo  multiplication of the form by non-zero complex scalars.
There is a natural map π(μ) → which factors through (μ) forgetting the differential.  We denote by (μ) the image of the stratum [g](μ) in  by π and by (μ) its closure in the Deligne-Mumford moduli space of stable curves . These loci are interesting  subloci of , but only a few results on them are known.  The dimension of (the irreducible components of) (μ) have been computed in <cit.> in the holomorphic case and <cit.> in the meromorphic case. Moreover, in the holomorphic case, when the locus (μ) is a divisor in , its class has been computed in <cit.>. 
In this paper, we study the loci [g](μ) in the meromorphic case. 
The main result of this article is the computation of the class of [3](6;-2) in the Picard group  ([3]) of the moduli stack [3]. Recall that this group is generated by the first Chern class λ of the Hodge bundle, and by the two boundary divisors δ_0 and δ_1.


 The class of [3](6;-2) in ([3])⊗ is

    [ [3](6;-2) ] = 17108λ- 1792δ_0 - 4396 δ_1 .


The computation of the  λ-class in ([3])⊗ is done in Section <ref> using Porteous formula. For the class in ([3])⊗ we use the theory of test curves and degeneration techniques in Section <ref>. This computation has been checked using sage package admcycles <cit.> based on a result of <cit.> proving a conjecture of Janda, Pandharipande,  Pixton and  Zvonkine. 
In order to perform the test curve in Section <ref>,  some enumerative problems related to abelian differentials naturally appear. In particular, we consider the differentials whose residue at every poles is equal to zero. These differentials are classically called differentials of the second kind. We denote by (μ) the locus of (μ) parametrizing the meromorphic differentials of type μ of the second kind. In Section <ref> we compute the dimension of the image of (μ)  by the forgetful map inside the moduli space of curves. 

Let μ = (a_1,…,a_n;-b_1,…,-b_p) be  a partition of 2g-2 such that p≥2 and b_i≥2 for all i. 

 
  * If g=1 the dimension of the projection of every component of [1](μ) to [1,1] forgetting all but one marked point is 1.
 
  * If g≥ 2, the dimension of the projection of (μ) to  is min{ 3g -3 ;2g + n - 2 }. 


After giving some general results on families of stable curves and multi-scale differentials in Section <ref>, we solve some enumerative problems on differentials of the second kind in  Section <ref>. 
The most interesting enumerative problem that we solve is the following one.

   The map π[1](6;-2,-2,-2) →[1,1] forgetting the polar points is an unramified cover of degree 7. 


This result can be interpreted in the following way. On a fixed curve X of genus 1 there exist 7 differentials in [1](6;-2,-2,-2) modulo translation on X and multiplication by ^∗ of the differential.

To conclude, note that according to <cit.> the strata of [3](6;-2) has three connected components. Each component gives rise to an irreducible component of the divisor [3](6;-2). We make some comments on this problem at the end of this paper (see Corollary <ref>) and in a future work we want to study each of these irreducible components.






  
Acknowledgements. 
We want to thank warmly Johannes Schmitt for detecting errors with the program <cit.> and for stimulating discussions leading to the corrected version.
We thank Dawei Chen, Noe Bárcenas and Scott Mullane  for their valuable help on several mathematical topics.  Moreover we thank Miguel Magaña Lemus and Gerardo Tejero Gómez for technical support.







§ BACKGROUND



In this section we recall some known facts, first about the multi-scale differentials  and then about Chern classes of the moduli space.
First we give  some common background  for both sections. Let consider the moduli space of smooth m-pointed genus g curves [g,m]. Let 
 π𝒳→[g,m] be the universal curve over [g,m], and let 
 ω_π:=ω_𝒳|[g,m] be the relative dualizing sheaf. We have that 
 Ωℳ_g,m:=π_*(ω_π) is a vector bundle over [g,m], that is the pull-back of the Hodge bundle.


Let n,p be strictly positive integers and we introduce the notation m = n+p. We denote by μ = (a_1,… ,a_n; -b_1,… ,-b_p) a m-tuple of integers such that a_i,b_j≥ 1 and ∑_j=1^n a_j - ∑_i=1^p b_i = 2g-2. 
We study meromorphic differentials η with zeroes of order a_i at z_i and poles of order b_j at w_j, i.e. such that (η)=(η)_0-(η)_∞=∑ a_j z_j-∑ b_i w_i. We will usually write  for the tuple of points (z_1,…,z_n;w_1,…,w_p).
For j=1,…, m, let 
 𝒟_j be the sections of the universal curve corresponding to the marked points z_j if j≤ n and w_n+j if j≥ n+1.
The strata Ωℳ_g(μ) of abelian differentials of type μ is defined to be the subspace of 
π_*(ω_π)(∑_i=1^p b_i𝒟_n+i) of differentials which vanish to order a_i at the sections 𝒟_i up to the action of the group permuting the singularities of the same order. 





 §.§ The multi-scale differentials



In this section we recall some notions that we need about twisted and multi-scale differentials as introduced in <cit.> and <cit.>.
We begin by recalling the notion of twisted differentials on which is based the more sophisticated notion of multi-scale differentials.

 A  ω of type μ (compatible with a full order )
on a stable n-pointed curve (X,) is a collection of
(possibly meromorphic) differentials ω_v on the irreducible components X_v of X  and a full order  on the set of these components,
such that no ω_v is identically zero, with the following properties:


  (0) (Vanishing as prescribed) Each differential ω_v is
holomorphic and non-zero outside of the nodes and marked points of X_v.
Moreover, if a marked point z_j lies on X_v, then _z_jω_v=m_j.

  (1) (Matching orders) For any node of X that identifies
q_1 ∈ X_v_1 with q_2 ∈ X_v_2,

    _q_1ω_v_1+_q_2ω_v_2 = -2 .


  (2) (Matching residues at simple poles)  If at a node of X
that identifies q_1 ∈ X_v_1 with q_2 ∈ X_v_2 the condition _q_1ω_v_1=
_q_2ω_v_2=-1 holds, then _q_1ω_v_1+_q_2ω_v_2=0.

  (3)(Partial order) If  a node of X  identifies
q_1 ∈ X_v_1 with q_2 ∈ X_v_2, then v_1≽  v_2 if and
only if _q_1ω_v_1≥ -1. Moreover,  v_1≍ v_2 if and only if
_q_1ω_v_1 = -1.

  (4) (Global residue condition) For every level i
and every connected component Y of X_>i that does not
contain a pole w_i the following condition holds.
Let q_1,…,q_b denote the set of all nodes where Y intersects X_(i). Then

    ∑_j=1^b_q_j^-ω=0 ,

where by definition q_j^-∈ X_(i).


Note that by point (1) of Definition <ref>, at each node of X the twisted differential ω, either has two simple poles or has a  zero of order k on one branch of the node and a pole of order -k-2 on the other branch. The prong number at a node is 0 in the first case and κ=k+1 in the second case. 
Now we give the definition of a multi-scale differential, referring to <cit.> for details.

 A multi-scale differential (X, , ω, , σ) of type μ is a stable pointed curve (X,) with =(z_1,…,z_n;w_1,…,w_p), a twisted differential ω of type μ over X compatible with the total order  and a global prong-matching σ.

The notion of prong-matching is introduced and discussed in great details in <cit.>. For us it will not be crucial to know its precise definition. It is sufficient to now that it gives a way to glue the differentials at the nodes. We will use mainly the facts that it is a finite data and that the number of possible classes of prong-matching is computable.  In the  important case of a multi-scale differential with two levels, this number is

    n  =  (κ_i)  ,

where i runs through the set of nodes of the multi-scale differential.



The importance of the notion of multi-scale differentials comes from the following theorem proved in <cit.>.

 The moduli space  of the multi-scale differentials of type μ is a  smooth compactification of the stratum (μ).


Moreover, there exists a good system of coordinates near the boundary of this space (see <cit.>). The perturbed periods coordinates give a way to understand the families of degenerating differentials with special properties. In this article, the relevant information that we need is that near a given multi-scale differential, the top differential is almost constant, while on the lower levels the differentials are multiplied by ∏ t_i^a_i, where t_i is a local parameter for each level of the multi-scale differential and a_i is an integer defined in <cit.>. Moreover, we know the local equations of the nodes in the universal family. 
In the useful case of a multi-scale differential with two levels we can be more specific. The local equation of the family at the node n_i is x_iy_i = t^a_i for a local parameter t and with a_i = (κ_n)/κ_i.








 §.§ Chern classes.



The goal of this section is to recall  some facts about Chern classes on the moduli space of curves. 
In order to make this section self contained, we begin by recalling some well-know facts of algebraic geometry.



  §.§.§ Some Notation.

 Let f Z↪ Y be a  a closed immersion of schemes, and denote the sheaf of Kähler differentials of Z over Y by Ω^1_Z|Y.
 We have the exact sequence

    0→ I_Z→𝒪_Y→𝒪_Y/I_Z=𝒪_Z → 0  ,

where I=I_Z is the ideal sheaf of Z. Let F be a coherent sheaf on Y, then tensoring the above exact sequence by F we get 

    0→ I_Z⊗ F→ F→ F⊗𝒪_Z=F_Z → 0 .

 The conormal sheaf  𝒩^∨_Z|Y of f is the quasi-coherent 𝒪_Z-module I/I^2 and the normal sheaf is 𝒩_Z|Y=Hom_𝒪_Z((I/I^2),𝒪_Z).   Suppose now that Z is an effective Cartier divisor and let F=𝒪_Y(Z) the associated invertible sheaf, then we have an exact sequence

    0→𝒪_Y→𝒪_Y(Z)→𝒪_Z(Z)→ 0  ,

 where 𝒪_Z(Z)=𝒪_Y(Z)|_Z. In this case we have that 𝒩^∨_Z|Y=𝒪_Y(-Z)|_Z and 𝒩_Z|Y=𝒪_Y(Z)|_Z. From the exact sequence

    0→𝒪_Y(-2Z)→𝒪_Y(-Z)→𝒪_Y(-Z)|_Z→ 0

 we have that 
 
    Ω^1_Z|Y≃ I/I^2=𝒪_Y(-Z)/𝒪_Y(-2Z)=𝒪_Y(-Z)|_Z=𝒩^∨_Z|Y

 and

    (Ω^1_Z|Y)^∨≃𝒪_Y(Z)/𝒪_Y=𝒪_Y(Z)|_Z=𝒩_Z|Y .

We recall that if A is a ring and N⊂ M⊂ L are A-modules, then L/M≃ (L/N)/(M/N). In our case we have the inclusion  𝒪_Y⊂𝒪_Y((n-1)Z)⊂𝒪_Y(nZ) of coherent sheaves on Y. Hence we can consider quotient sheaves to get the following exact sequences on Y

    0→𝒪_Y((n-1)Z)/𝒪_Y→𝒪_Y(nZ)/𝒪_Y→𝒪_Y(nZ)/𝒪_Y((n-1)Z)→ 0  .







  §.§.§ The setting.
 


Given any family π→ B of curves of genus g, we denote by ω_|B the relative dualizing sheaf of the family π. 
When the family π contains singular fibers, we have that ω_|B is equal to Ω^1_|B away of the nodes of the fibers, thus,  when the family is a family of smooth curves we can identify ω_|B≃Ω^1_|B.



Consider the universal curve π_0𝒳=ℳ_g,1→ℳ_g. For g≥ 2 this map is smooth of relative dimension one. We denote by 
Ω the relative dualizing sheaf ω_𝒳|ℳ_g associated to π_0. 
Let π^n𝒳^n → be the n-fold fiber product of 𝒳 over 
ℳ_g. The space 𝒳^n parametrises smooth genus g curves with n-tuples of not necessary distinct points.  Note that the fiber  over X∈ℳ_g of π^n is the direct product X^n= X×…× X and  the fiber of ℳ_g,n→ is the complement of the diagonal Δ in X^n, where 

    Δ={(X, q_1;…,q_n): q_i=q_j for at least two indices i≠ j} .

Let Δ_ij be the diagonal corresponding the points where q_i=q_j for two indices i≠ j. Let π_i𝒳^n→𝒳 be the forgetful map which forgets all but the i-th factor and let define the sheaf Ω_i=π_i^*(Ω). 






  §.§.§ The Chern classes for π_*(Ω_1(2Δ)).
 
We restrict our attention to the case n=1, that is, we consider the projection π=π_1𝒳^2→𝒳 on the first factor. We will write 
𝒪=𝒪_𝒳^2 and consider the diagonal of the fiber product on 
𝒳^2, that is, Δ𝒳→𝒳^2=𝒳×_ℳ_g𝒳, and we identify 
Δ(𝒳)≃𝒳. We write Δ to denote the image Δ(𝒳). From identifications in Equation (<ref>) and from the exact sequence (<ref>)  with Z=Δ and Y=𝒳^2, we have that the normal bundle 𝒩_Δ|𝒳^2 satisfies  

    𝒩_Δ|𝒳^2=𝒪_𝒳^2(Δ)/𝒪_𝒳^2≃(Ω^1_Δ|𝒳^2)^∨≃(π_1^*(Ω^1_𝒳|ℳ_g))^∨=(Ω_1)^∨=π_1^*(Ω)^∨ .

With this notation, we twist the sequence (<ref>) for n=2 by Ω_1=π_1^*(Ω) to get the exact sequence
 
    0→Ω_1⊗(Ω^1_Δ|𝒳^2)^∨→Ω_1⊗(𝒪(2Δ)/𝒪)→Ω_1⊗((Ω^1_Δ|𝒳^2)^2)^∨→ 0 .



 Using Equation (<ref>)  the previous exact sequence reads

    0→𝒪_𝒳^2→Ω_1⊗(𝒪(2Δ)/𝒪)→(Ω_1)^∨→ 0  .

 Pushing down this exact sequence to 𝒳 with π_* we have the following exact sequence coherent sheaves

    0→ F_0→ F_1→ F_2→ 0  ,

where F_0:=π_*(𝒪_𝒳^2)=𝒪_𝒳,  F_1:=π_*(Ω_1⊗(𝒪(2Δ)/𝒪)) and F_2:=π_*(Ω_1^∨). 
The fiber of F_1 at a point (X,p)∈𝒳 is the  two-dimensional vector space of sections H^0(X, K_X(2p)/K_X). Similarly the fibers of F_0  and F_2 at (X,p,q) are respectively  the vector spaces H^0(X,𝒪_X) and H^0(X,K_X^∨)={0}. So the exact sequence (<ref>) reads

    0→𝒪_𝒳→ F_1→π_*(Ω_1^∨)→ 0  ,

which implies that the Chern classes satisfy

    c(F_1)=c(𝒪_𝒳^2)c(π_*(Ω_1^∨))=1-K_1  ,

  where, following the notation in <cit.>, we have that K_1=c_1(π_*(Ω_1^∨)) is the first Chern class. 

 Let E=π_*(Ω_1) and E_2=π_*(Ω_1⊗𝒪_𝒳^2(2Δ))=π_*(Ω_1(2Δ)). The exact sequence 

    0→ E→ E_2→ F_1→ 0

implies that the Chern classes satisfy c(E_2)=c(E)c(F_1)=(1+λ)(1-K_1). 
This leads to the following result.

The Chern class of E_2 is
 
    c(E_2)= 1 + ∑_i≥1 (λ_i-λ_i-1K_1) .


To conclude, let us remark that we can define a similar vector bundle on ^n. It suffices to consider  the exact sequence 

    0→𝒪_𝒳^n→Ω_1⊗(𝒪(2Δ_1,n)/𝒪)→(Ω_1)^∨→ 0

and to define E_2 to be the push-forward by the map forgetting the last point of the middle term. Then Proposition <ref> remains true in this generalised context.


  §.§.§ The Chern classes for π_*(Ω_1(nΔ)) with n≥3.
 Now we want to extend the result of the formula of the Chern class of Proposition <ref> to the bundle E_n:=π_*(Ω_1(nΔ)) on 𝒳 for all n≥ 3. Note that the fiber of E_n   at a  point (X,p)∈𝒳 is the space H^0(X,K_X(np))  of differentials on X that have at worst poles of order n at p. 
By twisting the exact sequence (<ref>) by Ω_1 and using Equation (<ref>) we obtain the exact sequence

    0→π_*(Ω_1⊗𝒪((n-1)Δ)/𝒪))→π_*(Ω_1⊗𝒪(nΔ)/𝒪))→π_*((Ω_1^⊗ n-1)^∨)→ 0

on 𝒳.
This gives a recursive formula for Chern classes of E_n as follow

    c(π_*(Ω_1⊗𝒪(nΔ)/𝒪)))=c(π_*(Ω_1⊗𝒪((n-1)Δ)/𝒪))c(π_*((Ω_1^⊗ n-1)^∨)) .

Since the base case c(π_*(Ω_1⊗𝒪(2Δ)/𝒪)))=(1-K_1)   is given by equation (<ref>), we obtain 

    c(π_*(Ω_1⊗𝒪(nΔ)/𝒪)))=∏ _i=1^n-1 (1-iK_1) .



 Let X be a smooth curve and p∈ X. For all positive integer n we have the following exact sequence

    0→ K_X→ K_X(np)→ K_X(np)|_np→ 0  .

In order to globalize this exact sequence to the universal curve 𝒳 we proceed as follow. On 𝒳^2 we have the following exact sequence

    0→𝒪→𝒪(nΔ)→(𝒪(nΔ))|_nΔ→ 0 .

By tensoring with Ω_1 and pushing down to 𝒳 we obtain the following exact sequence

    0→π_*(Ω_1)→π_*(Ω_1(nΔ))→π_*(Ω_1(nΔ)|_nΔ)→ 0 .

Since the exact sequence (<ref>) gives that 𝒪_𝒳^2(nΔ)/𝒪_𝒳^2=𝒪(nΔ)|_nΔ, we identify the vector bundle π_*(Ω_1(nΔ)|_nΔ) with π_*(Ω_1⊗(𝒪(nΔ)/𝒪)))). Hence Equation (<ref>) gives the class c(E_n).
 
  The Chern class of E_n is
 
    c(E_n)= (1 + λ)∏ _i=1^n-1 (1-iK_1) .

 
Finally note that, as in the case of E_2, this proposition can be extended on ^n to the sheaf E_n   similarly defined.

 

 

  §.§.§ Known facts about Chern Classes.

 
 We conclude this section by recalling two known facts about Chern classes. The first one is the inversion formula for Chern classes. The second is some equalities for the Chern classes above the moduli space.
 We first give a formula in order to compute the inverse of a Chern class. Much more material around this circle of ideas can be found in <cit.>.
 Let us first define the polynomial 
 
    P_n (x_1,…,x_n) = ∑_i_1+2i_2+⋯+ni_n = n( (i_1+⋯+i_n)!/i_1!⋯ i_n!∏_j=1^n(-x_j)^i_j)  .

To be concrete, the polynomials P_n for n≤3  are 

    P_0   =    1  , 
    
    P_1   =    -x_1 , 
    
    P_2   =    x_1^2-x_2 , 
    
    P_3   =    -x_1^3 + 2x_1x_2 - x_3 .

The importance of these polynomials is given by the following result that will be used several times in Section <ref>.

 Let E be a complex vector bundle over a complex manifold X whose  Chern class is c(E)=1+c_1(E)+ c_2(E) + ⋯ + c_n(E). Then the Chern class of -E is 
 
    c(-E)=1+P_1(c_1)+P_2(c_1,c_2)+⋯+ P_n(c_1,…,c_n) .


We now recall some results about Chern classes on the moduli space. These results are due to <cit.> and many examples of application can be find in <cit.>.

 Let K_i be the class of Ω_i = π_i^*(Ω) and Δ_ij be (the class of) the  (i,j)-th diagonal as introduced in Section <ref>, then
 
    Δ_idΔ_jd   =   Δ_ijΔ_id for  i<j<d ,
    Δ_ij^2   =   -K_iΔ_ij for  i<j ,
    
     K_jΔ_ij   =   K_iΔ_ij for  i<j .

Moreover denoting by π𝒳^d-1→ the forgetful map, then for every monomial M pulled back from 𝒳^d-1
 
    π_d,∗(M·Δ_id)    =    M ,
    π_d,∗(M· K_d^k)    =    M ·π^∗(κ_k-1) .












§ THE LOCUS OF DIFFERENTIALS OF THE SECOND KIND




The objective of this section is to compute the dimension of the projection to  of the locus parametrizing the differentials of second kind in the strata of meromorphic differentials. Recall that the differentials of second kind are meromorphic differentials such that the residue at every pole is zero. We denote the locus of differentials of type μ of second kind of type μ by (μ).
We begin with a preliminary result.

Given a stratum of meromorphic differentials (a_1,…,a_n;-b_1,…,-b_p) such that g≥1, p≥2 and b_i≠ 1. The subspace which parametrises differentials with zero residues at the first i≤ p-1 points is of codimension i. 





By <cit.> these subspaces are not empty. Moreover, each condition on the residues gives a linear equation  in period coordinates. Since these equations involve at most p-1 residues, the i equations are independent. Hence the locus that they define is of codimension i.


We can now compute the dimension of the projection of the loci of differentials of second kind to , proving Theorem <ref>.  Recall that this theorem says that the dimension of the projection is 1 in the genus one case and min{ 3g -3 ;2g + n - 2 } in the genus g≥2 case.
The proof is by degeneration in the spirit of Theorem 1.3 of <cit.>. In this proof, we use the notation b=∑_i=1^pb_i.


For g=1, if there were a component of [1](μ) such that the dimension of the projection 0, then this implies that there exists  a curve X having a one dimensional family of differentials of the second kind of  type μ on X. We know (see for example Section 2 of <cit.>) that a differential of type μ on X can be written 

    ω = σ^a_1(z-z_1)…σ^a_n(z-z_n)/σ^b_1(z-w_1)…σ^b_n(w-z_n) dz   ,

where σ is sigma-fonction of X. Hence the residue is a rational function of the z_i and w_j which is clearly non constant.
For g≥ 2, we degenerate to the curve X pictured in Figure <ref>. The curve X_0 is of genus g-1 and the genus of X_1 is 1. We denote by p_0 and p_1 the nodal points belonging respectively to X_0 and X_1.
 

We consider the twisted differential ω on X such that the restriction ω_i to the irreducible component X_i are the following differentials. On X_0 the differential is in the stratum [g-1](a_1,…,a_n;-b-2) and on X_1 the differential is in [1](b;-b_1,…,-b_p). Note that by <cit.> this twisted differential is in the closure of (μ). 
Moreover by <cit.> the dimension of the projection [g-1](a_1,…,a_n;-b-2) to [g-1] is min(3(g-1)-3,2(g-1)-2+n).
Suppose that n≥ g-1, then  there exist a dense subset U of [g-1] such that for every point X'_0 in U there exists a differential ω_0 on X'_0. Note moreover that there is a positive dimensional family of such differentials with the polar point p_0 moving on X'_0.
By the case of genus g=1, the dimension of the projection of [1](b;-b_1,…,-b_p) is equal to one. Moreover, there is one smoothing parameter at the node.
Summing up the contribution, the dimension of the projection is 3(g-1)-3 + 1 +1 +1 = 3g-3, where 3(g-1)-3 is the dimension of the projection of [g-1](a_1,…,a_n;-b-2) to [g-1] and the three 1 are respectively the moving point p_0∈ X'_0, the dimension of the space of elliptic curves {(X_1,p_1)} and the smoothing parameter of the node. 
If n < g-1 then by <cit.> there exists a  2(g-1)-2+n dimension space of curves X_0 which admits such differential ω_0. Moreover the set of possible point p_0 is finite on X_0. Hence the dimension of the projection of the locus (μ) is 2(g-1)-2+n +1 +1 =2g-2+n in this case. In this sum, the contribution 2(g-1)-2+n is the dimension of the projection of 
[g-1](a_1,…,a_n;-b-2) to [g-1] and the two 1 are respectively the dimension of the space of elliptic curves { (X_1,p_1) } and the smoothing parameter of the node.


 
 
 



§ THE CLASS OF THE DIVISOR ℳ_3(6;-2)
 


The goal of this section is to compute the class of the projection [3](6;-2) of  [3](6;-2) to [3]. This gives the coefficient of  the λ-class in Equation (<ref>) of Theorem <ref>. In order to do this, we use Porteus Formula and the results recalled in Section <ref>.


For n≥ 1, set 𝒪=𝒪_𝒳^n. 


Recall that the diagonal Δ_ij is given by the locus where q_i=q_j in 𝒳^3 and define the divisor  
𝔇_2^6 = 2Δ_23-6Δ_13  inside 
𝒳^3. Tensoring the exact sequence

    0→𝒪(-6Δ_13)→𝒪→𝒪|_6Δ_13→ 0

by Ω_3(2Δ_23) we obtain

    0→Ω_3⊗𝒪(-6Δ_13+2Δ_23)→Ω_3⊗𝒪(2Δ_23)→Ω_3⊗𝒪(2Δ_23)|_6Δ_13→ 0 .

Pushing down this exact sequence  on 𝒳^2 by the map π forgetting the third point, we obtain

    0→π_*(Ω_3(𝔇_2^6))→π_*(Ω_3⊗𝒪(2Δ_23))→π_*(Ω_3(2Δ_23)|_6Δ_13)→ 0  .

 We define the following coherent sheaf on 𝒳^2

    ℱ=π_*(Ω_3⊗(𝒪(2Δ_23)/(𝒪(𝔇_2^6)))   .
 
Note that the stalk ℱ_x of ℱ at a point x=(X;z,w)∈𝒳^2 is given by

    ℱ_x=H^0(X,K_X(2w)/K_X(2w-6z)) .

The evaluation map gives a morphism ϕ E_2→ℱ over 𝒳^2 such that on the stalk we have

    ϕ_x  (E_2)_x=H^0(X,K_X(2w))→ℱ_x=H^0(X,K_X(2w)/K_X(2w-6z)) .

Since the dimension of the source is 4 and the dimension of the goal is 6, by Porteous formula (see <cit.>) the class of the degeneracy locus where the map has rank less or equal to three is c_3(ℱ-E_2). Note that this degeneracy locus can contain extra components in the diagonal Δ_1,2 = { (w,z): w=z }. This is indeed the case and we will deal with this problem at the end of this section.
The Chern classes of ℱ is given by the following general formula.

 The Chern class of the vector bundle whose fiber is
 
    H^0( K(∑_i=1^p b_iw_i)/K(∑_i=1^p b_iw_i-∑_i=1^n a_iz_i)  )

 is equal to
 
    ∏_i=1^n∏_j=1^a_i (1+jK_p+i+∑_k=1^p b_kΔ_k,p+i)  .




Given an effective divisor D on a smooth curve X, 
the sheaf K_X(D) is the sheaf of meromorphic 1-forms ω such that for a point q on X,  
(div ω)(q)+D(q)≥ 0, in particular if q∉supp(D), then ω is holomorphic at q. If q is a zero of order a for ω, then in an open neighbourhood U around q with local coordinate z, we can trivialize K_X so that the form z^a·ω generates the stalk K_X(D)|_q, then K_X(D)|_q≃ K_X^⊗ a|_q⊗𝒪_q(D). 
Recall that we denote by μ=(a_1,…,a_n;-b_1,…,-b_p) a partition of 2g-2  where a_i,b_j≥ 1 and that m=n+p. 

On 𝒳^m we denote Δ_Z=∑_i=1^n a_iΔ_p+i,m+1 and Δ_W=∑_j=1^p b_jΔ_j,m+1 the divisor of zeros and poles respectively. We then consider the divisor Δ_Z,W=Δ_Z-Δ_W
and the following two exact sequences on 𝒳^m+1 and 𝒳^m respectively:

    0→𝒪(Δ_W-Δ_Z)→𝒪(Δ_W)→𝒪(Δ_W)|_Δ_Z→ 0   ,


    0→π_*(Ω_m+1(𝒪(Δ_W-Δ_Z)))→π_*(Ω_m+1(𝒪(Δ_W)))→π_*(Ω_m+1(𝒪(Δ_W)|_Δ_Z))→ 0 .

 We recall that 𝒪(Δ_W)|_Δ_Z=𝒪_Δ_Z(Δ_W)=𝒪(Δ_W)⊗𝒪_Δ_Z. We set 

    ℱ:=π_*(Ω_m+1(𝒪(Δ_W)|_Δ_Z))=π_*(Ω_m+1(𝒪(Δ_W))/𝒪(Δ_W-Δ_Z))≃π_*(Ω_m+1(Δ_W)/Ω_m+1(Δ_W-Δ_Z)) .

 To compute the Chern classes of ℱ we adapt a classical argument that we learned in  <cit.>. 
 For every i=1,…,n, we consider  the bundle Ω_m+1(Δ_W-a_iΔ_p+i,m+1)=Ω_m+1⊗𝒪(Δ_W-a_iΔ_p+i,m+1)  on 𝒳^m+1, where Ω_m+1 is Hodge bundle on 𝒳^m+1. 
Consider the case i=1 and in order to simplify the notation we will use  for the diagonal Δ_p+1,m+1.  We have exact sequences:

    0→Ω_m+1(Δ_W-a_1)→Ω_m+1(Δ_W-(a_1-1))→Ω_m+1(Δ_W-a_1)|_→ 0 ,


    0→Ω_m+1(Δ_W-a_1)→Ω_m+1(Δ_W)→Ω_m+1(Δ_W)|_a_1→ 0  ,


    0→Ω_m+1(Δ_W-(a_1-1))→Ω_m+1(Δ_W)→Ω_m+1(Δ_W)|_(a_1-1)→ 0 .


We define 

    F_1:=π_*(Ω_m+1(Δ_W-(a_1-1))/Ω_m+1(Δ_W-a_1)),
        F_0:=π_*(Ω_m+1(Δ_W)/Ω_m+1(Δ_W-a_1)),
        and  F_2:=π_*(Ω_m+1(Δ_W)/Ω_m+1(Δ_W-(a_1-1))) .

From the natural inclusions 

    Ω_m+1(Δ_W-a_1)⊂Ω_m+1(Δ_W-a_1)⊂Ω_m+1(Δ_W-(a_1-1))⊂Ω_m+1(Δ_W)
 
we have the exact sequence 

    0→ F_1→ F_0→ F_2→ 0 .

Since the divisor  has multiplicity a_1 at Δ_Z and ∉Supp(Δ_W), the fiber of F_1 at z_1 is 

    (K_X(∑_j=1^p b_jw_j ))|_z_1≃ (K_X|_z_1)^⊗ a_1⊗𝒪_z_1(∑_j=1^p b_jw_j ) ,
 
then we have that 
F_1≃Ω_n^⊗ a_1⊗𝒪_𝒳^n(Δ_W). 
Following this argument we can to construct a new filtration 

    0→ F'_1→ F_1→ F'_2→ 0  ,
 
 where F'_1 is the sheaf defined as  
 
    F'_1:=π_*(Ω_m+1(Δ_W-(m_2-1))/Ω_m+1(Δ_W-(a_1-1))) ,
 
 and we have that F'_1≃Ω_n^a_1-1⊗𝒪_𝒳^n(Δ_W). In this way using a sequence of filtrations obtained by subtracting 1 successively to a_1 we get that 

    F_0≃[Ω^⊗ a_1_n⊗𝒪_𝒳^n(Δ_W)]⊗[Ω^⊗ (a_1-1)_n⊗𝒪_𝒳^n(Δ_W)] ⊗…⊗[Ω_n⊗𝒪_𝒳^n(Δ_W)]   .
 
By subtracting 1 successively to all a_i we can reduce to signature (1,0,…,0; -b_1,…,-b_p) to obtain the expression for the Chern class of ℱ as desired.




Applying Lemma <ref> to the case of the stratum [3](6;-2), we get that

    c(ℱ)    =   ∏_j=1^6 (1+jK_2+2Δ_12)  ,
       =    1+ (21K_2+12Δ_12) +(175 K_2^2+210K_2Δ_12+60Δ_12^2)
           + (735K_2^3+1400K_2^2Δ_12+840 K_2Δ_12^2+160 Δ_12^3) + … .

By Proposition <ref> we obtain that the class of -E_2 is 

    c(-E_2)    =    1+(K_1-λ_1) +(K_1^2-λ_1K_1 +λ_1^2-λ_2) 
           + (K_1^3-λ_1K_1^2 +(λ_1^2-λ_2)K_1 +2λ_1λ_2  -λ_3 -λ_1^3 ) + … .

So we obtain 

    c_3(ℱ-E_2)    =    (735K_2^3+1400K_2^2Δ_12+840 K_2Δ_12^2+160 Δ_12^3)
           + (175 K_2^2+210K_2Δ_12+60Δ_12^2)· (K_1-λ_1)
           + (21K_2+12Δ_12)· (K_1^2-λ_1K_1 +λ_1^2-λ_2) 
           + (K_1^3-λ_1K_1^2 +(λ_1^2-λ_2)K_1 +2λ_1λ_2  -λ_3 -λ_1^3 ) .

This is equal to the Chern class of ℳ_3(6;-2) inside A^3(ℳ_3,2). We now simplify this expression. We first expand this sum, leading to the equality

    c_3(ℱ-E_2)    =    735K_2^3 + 175(K_1 - λ_1)K_2^2   + 21 (K_1^2 -  λ_1 K_1 + λ_1^2- λ_2) K_2
            + 1400 Δ_12 K_2^2  + 210 Δ_12 K_1 K_2   - 210 Δ_12λ_1 K_2 + 840 Δ_12^2  K_2
            + K_1^3  - λ_1 K_1^2  + 12 Δ_12 K_1^2   + (λ_1^2 - λ_2) K_1 - 12 Δ_12λ_1 K_1 + 60 Δ_12^2  K_1 - λ_3
            + 2 λ_1λ_2 - 12 Δ_12λ_2 - λ_1^3 + 12 Δ_12λ_1^2  - 60 Δ_12^2λ_1 + 160 Δ_12^3 .


We now simplify this expression using the equalities recalled in the first part of Lemma <ref>. For example, the second equality of this lemma implies that the coefficient 60 Δ_12^2  K_1 is equal to -60Δ_12K_1^2.
Doing this for all the  coefficient, this gives the expression 

    c_3(ℱ-E_2)    =    735K_2^3 + 175(K_1 - λ_1)K_2^2+ 21 (K_1^2 -  λ_1 K_1 + λ_1^2- λ_2)  K_2
           + (882K_1^2-162λ_1K_1+12(λ_1^2-λ_2))Δ_12
           + K_1^3-λ_1K_1^2+(λ_1^2-λ_2)K_1 - λ_3+ 2 λ_1λ_2-λ_1^3 .

We now forget the two marked points to obtain the λ-class of ℳ_3(6;-2) in the Picard group of ℳ_3.
By forgetting the second point and using the formulas of the second part of Lemma <ref> we obtain

    π_2,∗( c_3(ℱ-E(2)) )     =    735κ_2 + 175(K_1 - λ_1)κ_1+ 21 (K_1^2 -  λ_1 K_1 + λ_1^2- λ_2)  κ_0
           + 882K_1^2-162λ_1K_1+12λ_1^2-12λ_2 .

And finally by forgetting the first point we get

    π_1,∗∘π_2,∗( c_3(ℱ-E(2)) ) =  882κ_1-162κ_0λ_1 + 175κ_0κ_1 -21κ_0^2λ_1 +21κ_0κ_1 .

Hence using that κ_0=4 and κ_1= 12λ_1 we obtain that the class of π_1,∗∘π_2,∗( c_3(ℱ-E(2)) ) in ([3]) is

    [  π_1,∗∘π_2,∗( c_3(ℱ-E(2)) ) ] =  19008 λ_1 .







The degeneracy locus of the map ϕ E_2→ contains the locus that we are interested in of curves X together with points w and z such that there exists a differential with divisor 6z - 2w and a locus supported in the diagonal Δ. So in order to obtain the class of [3](6;-2) in the Picard group of [3] it remains to substract this contribution. This is in general a delicate task and we refer to <cit.> for some examples.
At a point (X,z,z) of the diagonal, the evaluation map restricts to 

    ϕ_z H^0(X,K_X(2z))→ H^0(X,K_X(2z)/K_X(-4z))  .

This has generically rank 4 as expected and has rank 3 exactly if there is an abelian differential on X which has a zero of order 4 at z.  The class of this locus is classical but we recall it for completeness.
We have

    c() = 1+ 10 K_1 +35 K_1^2  + ⋯ .

Hence 

    c_2( - E) = λ_1^2 - 10 λ_1 K_1 +35 K_1^2 .

When pushing down to [3] we obtain

    π_1,∗( c_2( - E))    =    35 κ_1 -10 λ_1κ_0
       =    35 · 12λ_1 - 10 · 4 λ_1
       =    380 λ_1 ,

where we used again that κ_1 = 12 λ_1 and κ_0 = 4.

Now we compute the multiplicity of this locus. Consider a point (X,z) where z is such that there exists a differential ω on X whose divisor  is 4z. Locally the vector bundle E_2 is generated by four sections. Let u is a local coordinate of X at z, we can take these sections to be η_t=duu^2, the family of  differentials u^2η_t, a family of differentials vanishing at order 1 or 2 at z depending if X is non hyperelliptic or hyperelliptic respectively and finally a family of the form ω_t := u^2(u^4+s)ω_0 where s is a function of the base vanishing at X. So the locus of pointed curves (X,z) is given by the scheme given by the two by two  minors of the matrix

    (
      [          1        u^2       u^3h u^2(u^4+s);          0         2u ∂_u( u^3h)  6u^5+2u s ]) ,

where h depends if the point is or not. It is easy to check that the column where it appears the function h will  not contribute to the multiplicity. In fact, the multiplicity of this point is given by the order of vanishing of s. In order to compute this order note that ω_t leads to a family of of multi-scale differentials after blowing up the diagonal Δ. Note that the equation of the node is then given by xy=f where f is the function defining the diagonal. Moreover, the function s is a rescaling parameter for this family of multi-scale differentials as defined in <cit.>. More precisely, according to Equation (7.1) and Definition 7.2 of <cit.>, we have s=f^5, where the 5 is the prong number. This implies that  the multiplicity of the considered locus is 5.

Summing up all the computations of this section, we obtain that the class of  [3](6;-2) in the Picard group of [3] is 

    [ [3](6;-2) ] =  19008 λ_1 - 5 · 380 λ_1  = 17108λ_1 .




























§ CLASS OF THE DIVISOR [3](6;-2)



In this section we compute the class of the locus [3](6;-2)  in the rational Picard group ([3])⊗, proving Theorem <ref>. We first give in Section <ref> some general facts on degenerating families of curves and differentials. Then we solve in Section <ref> some enumerative problems related to differentials of the second kind. And finally in Section <ref> we use this information to compute the class [[3](6;-2)] using the method of test curves. 











 §.§ General facts on degenerating families




First we consider the problem of finding out how many fibres are isomorphic in a family of semi-stable curves. This generalizes a result of <cit.> that we could not find in the literature in the following general form.


  Let f→ (B,p) be a family of generically smooth semi-stable curves over a smooth curve such that the fibre X_p has exactly one unstable rational component E. We denote by n_i the two nodes of E.  Given   a local parameter t of B at p, if the family is of the form x_1y_1=t^a_1 and x_2y_2=ct^a_2 with |c| =1  at the nodes n_i,  then the induced application μ_f B → is a  cover ramified at p of degree  a_1+a_2 on its image. 
 
 
 
 
 The proof has two parts.   First we are going to give a condition so that two curves are isomorphic near X_p. Then we are going to count the number of curves that satisfy that criterion in the hypotheses of the lemma. 
 Let X_p be a semi-stable curve as in Lemma <ref> with  exceptional component E. Consider the family →Δ^2 = { (t_1,t_2) : |t_i| < 1 } which is given locally at the nodes by the equations x_iy_i=t_i where x_i are local equations of E in  and y_1y_2=1. Consider the two curves X_θ_1,θ_2 and X_ϑ_1,ϑ_2 above the parameters (r_1exp(iθ_1),r_2exp(iθ_2)) and (r_1exp(iϑ_1),r_2exp(iϑ_2)). We now show that these curves are isomorphic if and only if θ_1+θ_2=ϑ_1+ϑ_2. 
 Let us study when the identity 
 of a neighborhood of a complement of x_i=0 in the curves X_θ_1,θ_2 and X_ϑ_1,ϑ_2 can be extend to an isomorphism φ X_θ_1,θ_2→ X_ϑ_1,ϑ_2. The identity can be extended through the annuli x_iy_i=t_i by the functions 
 
    ϕ_i (x_i,y_i) ↦ (x_i,exp(i(θ_i-ϑ_i))y_i) .

The functions ϕ_i can be extended to an holomorphic fonction on E if and only if ϕ_1 coincides with ϕ_2 on E, that is 

    exp(i(θ_1-ϑ_1))y_1 = exp(-i(θ_2-ϑ_2))y_2 = exp(-i(θ_2-ϑ_2))y_1^-1 .

  From that equation it follows directly that the curves X_θ_1,θ_2 y X_ϑ_1,ϑ_2 are isomorphic if  and only if θ_1-ϑ_1+θ_2-ϑ_2=0.
 Now we fix two integers a_1,a_2≥1 as in lemma <ref> and a real number r>0. We introduce the curve 
 
    _a_1,a_2,c = { (a_1θ,c+a_2θ) : θ∈𝕊^1}⊂𝕊^1×𝕊^1  .

It follows from the first part of the proof that it suffices to show that  the antidiagonal = { (θ,-θ) : θ∈𝕊^1} in 𝕊^1×𝕊^1  and the locus _a_1,a_2,c have a_1+a_2 points of intersection. Since for every c the curve _a_1,a_2,c is a translation of _a_1,a_2,0 the number of intersections does not depend on c.  Assume first that a_1 and a_2 are prime to each other. The intersections between  and _a_1,a_2,0  are given by the points (k/(a_1+a_2),-k/(a_1+a_2)) with k∈{0,…,a_1+a_2-1}. Hence there are a_1+a_2 points of intersection between  the loci  and _a_1,a_2,0.
 If  d=(a_1,a_2)>1 then the curves  and _a_1,a_2,0 have  a_1/d + a_2/d points of intersection, each of multiplicity d.
 



 The following result will be very useful in order to compute some intersection numbers in Section <ref>. 

 Given a twisted differential (X;z,w;ω) of type (a;-b) of  genus g≥ 2. The pointed curve (X;z,w) is not one of the curves pictured in Figure <ref> where E is a rational curve.  
  


Before doing the proof, we want to remark that it seems to be a general principle that special points related to differentials do not easily converge to a separating node. On the other hand, we will see in Section <ref> that special points can easily converge to separating nodes. For an other instance of this principles, the reader can look at  <cit.>.


In these three cases the restriction of ω to E has at least two poles. Hence Theorem 1.5 of <cit.> implies that the residues of these nodes are different from zero. Hence the global residue condition, that is recalled in Definition <ref>, can not be satisfied on any of these curves.


Finally, we give the form of the curve underlying a multi-scale differential of second kind with a unique zero in genus 1.


 The  multi-scale differential (X;z,w_1,…,w_p,ω;σ;) in the boundary of the locus [1](a;-b_1,…,-b_p) of differentials of the second kind are such that X is either irreducible (with one node) or is two rational curves which are glued together at two nodes.




If X was a curve not of one type given in Lemma <ref>, then either X has a rational component with marked points glued at the rest of the curve at one node or it is a chain of rational components  of length strictly greater than 2.
In the first case the differential on the projective curve has exactly one zero. Hence by <cit.> it has some poles with non zero residues. This implies that this differential can not be in the limit of a family of differential of [1](a;-b_1,…,-b_p) as proved in <cit.>.
In the second case, consider the component X_1 of X which contains the zero z of the multi-scale differential. Then the component X_1 is the unique local minimum for the full order  as shown in <cit.>. This implies that the differential ω|_X_1 has two poles at the nodal points of X_1. On the adjacent components X_2 and X_3 the differentials ω|_X_i have a zero at the corresponding nodal point. Hence at the other nodal points the differentials on ω|_X_2 and ω|_X_3 have another zero. This implies that there are both higher for  than some other components of X. Hence there is another local minimum, contradicting the unicity of the local minimum for .








 §.§ Some enumerative problems




 
Let us begin with a proposition giving the number of differentials with a unique zero and a unique pole on a curve of genus 2. This is essentially known (see  <cit.> or <cit.>) but we recall since it is only implicitly stated in the cited articles.

For every a≥ 2, the application π[2](a+2;-a)→[2] is an  unramified cover of degree 2(a+2)^2a^2-18.





 Let C be a general curve of genus 2 and consider the map 
    f C^2→^2(C) : (p_1,p_2) ↦ (a+2)p_1 -a p_2 .
 By <cit.> and <cit.> we know that f is a finite map of degree  2(a+2)^2a^2. Moreover, this application is ramified at the loci
 
    Δ = ( (p_1,p_2)∈ C^2 : p_1=p_2)

and 
 
    = ( (p_1,p_2)∈ C^2 : p_1=ι(p_2) ) ,

 where ι is the hyperelliptic involution. Note that Δ∩ is the set of the six points of C.  Let us consider the preimage by f of the canonical bundle of C. If a pair (p_1,p_2) is in the preimage of K_C belongs to Δ, then (a+2-a)p_1 is canonical, so p_1 is a point. Second, if a pair (p_1,p_2)  of the preimage of K_C belongs to , then there are differentials ω_1 and ω_2 whose respective divisors are (a+2)p_1-ap_2 and p_1+p_2. Then η = ω_1ω_2^a is a (a+1)-differential with a unique zero of order 2a+2. The (a+1)-differential η exists on a general curve if and only if η is the (a+1)th power of an abelian differential with a double zero. This implies that p_1=p_2 is a point. 
 We now compute the degree of ramification of f at the points as in <cit.>. Let p be a point and (ω_1,ω_2) generators of H^0(C,K_C) such that _p(ω_1)=0 and _p(ω_2)=2. The derivative of f at p is of the form
 
    Df(p)=[     1     1; t_1^2 t_2^2 ]

where t_i are functions which vanish at p at order 1. Hence the determinant of this matrix vanishes at order 2 at p. It follows that the order of branching at p is equal to 2. 
Hence there are 18 pairs in the preimage of the canonical divisor K_C that do not correspond to solutions in the stratum [2](a+2;-a) but to abelian differentials with a single zero order 2. 
It remains to show that the cover is not branched. Suppose it is branched, then the branching point corresponds to a twisted differential where the underlying curve is a genus 2 curve with a rational curve E glued at a point. The curve E contains the zero of order a+2 and the pole of order -a. Hence the restriction of the differential to E has two poles and a unique zero, hence the residues at the poles are non-zero.  Hence this twisted differential does not satisfy the global residual condition of Definition <ref> and is not smoothable.

We will now deduce an important consequence of this result for the locus of differentials of second kind on elliptic curves. 


 The degree of the map π[1](a+2;-a,-2)→[1,1] forgetting the pole is equal to (a+2)^2 + a^2 -10 for a≥3 and 5 for a=2.



Let us consider a stable curve X of genus 2 which is the union of two elliptic curves attached at one node. We describe all the multi-scale differentials on a curve X semi-stably equivalent to X. By Lemma <ref> the pointed curves which can appear in such multi-scale differentials  are of the form pictured in Figure <ref>. 

In case (a) the differential ω_1 on X_1 is in the stratum [1](a+2;-a-2) and the differential ω_2 on X_2 is in the stratum [1](a;-a). Hence there are (a+2)^2-1 different differentials on X_1 and a^2-1 different differentials on X_2. Since there is clearly only one class of prong-matching there are ((a+2)^2-1)(a^2-1) different multi-scale differentials of this type. Moreover, the space of multi-scale differentials is smooth at this point, hence  ((a+2)^2-1)(a^2-1) degenerate to this type. The type (b) is similar. Hence the intersection number with the ones of type (c) (or (d)) is equal to

    D    =     ((a+2)^2a^2 -9) - ( ((a+2)^2-1)(a^2 -1) )  ,
       =    2a^2+4a -6 ,
       =    (a+2)^2+a^2-10 .

Since the space of multi-scale differentials is smooth at these points and there is a unique differential on X_2 (in the case (c)), this implies that the number of distinct differentials of the second kind in [1](a+2;-a,-2) is (a+2)^2 + a^2 -10 for a≥3. The case a=2 is similar taking into account that there is a symmetry since both poles have the same order.

We now compute the degree of the map π[1](6;-2,-2,-2) →[1,1] proving Theorem <ref>. In order to illustrate the ideas of the proof in easier context, we first compute the degrees of the maps π[1](4;-2,-2) →[1,1] and π[1](5;-3,-2) →[1,1] (that is already known by Corollary <ref>).
 
 The degree of the map π[1](4;-2,-2) →[1,1] forgetting the poles  is 5 and the degree of the application π[1](5;-3,-2) →[1,1] is 42. 


The proof of this result is by degeneration in the moduli space of multi-scale differentials. The results that we use here are recalled in Section <ref>.


 We first consider the case of the locus [1](4;-2,-2) of differentials of the second kind of type (4;-2,-2).  By Lemma <ref> there are 3 types of pointed curves on which there can exists a multi-scale differential in the closure of this locus. This is summarised in Table <ref> that we first explain in details. Note that Tables <ref> and <ref> have the same entries.
 
 The first line gives the pointed stable curves on which there may exist a multi-scale differential. This is only given up to permuting the poles of the same order, for example in the second column, the case where w_1 and w_2 are permuted gives the same differential. The numbers κ_i are the prong numbers at the corresponding node.  
 The second line gives the number of twisted differentials that exit on the pointed curve under consideration. 
 The third line gives the number of classes of prong-matching. In the cases we will consider here, the multi-scale differentials have only two nodes and two levels. Hence this number is simply given by (κ_1,κ_2).
 The fourth line gives the local degree of  the stabilisation map [1,m] →[1,1] restricted to the smooth pointed curves underlying the differentials obtained by smoothing the multi-scale differential under consideration. This degree is given by Lemma <ref>. Note that if there is more than one class of prong-matching, the degree is given for each choice of prong-matching but it does not depend on this choice.
 According to the fourth line, on a smooth curve near the boundary of [1,1] the number of pointed differentials on this curves which degererate to the multi-scale differential under consideration is equal to the degree. However, it happens that these pointed differentials only differ by a permutation of the marked points.
 The fifth line gives the order of the group generated by these permutations of the marked points. 
 Finally in the last line we group all these informations to give the number of differentials of  type μ on a smooth curve which degenerate to a multi-scale differential such that the underlying pointed curve is the one of the first line.
 We now come back to the case of [1](4;-2,-2).
 Consider the irreducible pointed curve pictured in the first column. Any multi-scale differential on this pointed curve has two single poles at the node. Hence the pull-back of the differential on the normalisation of the curve  lies in the stratum  [0] (4;-2,-2,-1,-1) and the residues at the poles of order 2 are equal to 0. There is a unique such differential which is shown in Figure <ref>. 

Moreover, the number of prong matching and the local degree are clearly equal to 1. Hence on a smooth curve near the boundary of [1,1] there is a unique differential which degenerate to a multi-scale differential with this underlying pointed curve.
 We now consider the curve pictured in the second column of Table <ref>. First note that the prong numbers have to be equal to 1, since otherwise there would exist no full order . Hence the differential on the upper component is in the stratum [0](0,0;-2) and the differential on the lower component is in the stratum  [0](4;-2,-2,-2) and the residue vanishes at the marked pole. There is clearly a unique differential in [0](0,0;-2). Moreover <cit.> show that there is a unique differential satisfying the second conditions which is shown in Figure <ref>.  
 
 Since there is a unique class of prong-matching, there is a unique multi-scale differential ω with this underlying pointed curve. By Lemma <ref> in the case a_1=a_2=1 there exists a neighborhood of the boundary of  [1,1] on which there are two differentials near ω. 
 Moreover, as recalled in Section <ref>, in a neighborhood of this multi-scale differential the restriction of the differential on the top component looks like ω_0 and on the bottom component like tω_-1. The curves are isomorphic for the parameters t and -t and hence  the differentials on these curves are distinct.
We now consider the case pictured in the third column. Since the residues of the poles of order -2 vanish, Theorem 1.5 of <cit.> implies that κ_1=κ_2=2. By proposition 2.3 of <cit.> there is a unique element in [0](1,1;-2,-2) such that the residues at the poles vanish. Moreover there are 2 classes of prong-matching which give two disjoint families of multi-scale differentials. 
Locally at the nodes n_i of the pointed curve, the first family _1 is given by x_iy_i=t  and the second one _2 is given by x_iy_i = (-1)^it for i=1,2.  
Moreover we know that in a neighborhood of the multi-scale differential the form on the top component looks like ω_0 and on the bottom component like t^2ω_-1. Hence the differentials for t and for -t are equals to each over. Hence on the isomorphic curves above t and -t both differentials are isomorphic. Hence there is a unique differential degenerating to each multi-scale differential on this pointed curve.
We conclude that the number of distinct differentials in [1](4;-2,-2) on a smooth curve of genus 1 is 1+2+2=5.
We now deal with the case of the locus [1](5;-3,-2). There are four different types of pointed curves on which there can exist a multi-scale differential in the limit of the locus [1](5;-3,-2). These pointed curves are pictured in the first line of Table <ref>.
  
 Consider the irreducible curve of the first column. In this case the differential has simple poles at the nodal points. So its pull-back lies in the stratum [1](5;-3,-2,-1,-1) and its residues at the non simple poles vanish. 
 By Proposition 3.8 of <cit.> there are precisely 2 non isomorphic such differentials represented in Figure <ref>. 
 
 Moreover the simple poles can be distinguished by the fact that one is next to the pole of order -2 and the other next to the pole of order -3. Hence we obtain 4 such multi-scale differentials. It is easy to see that each of these differentials can be smoothed in a unique way.
 Now consider the curve in the second column of Table <ref>, when the pole of order -2 is on the top component. The differential on the top component is the unique differential in [0](0,0;-2). The  differential on the bottom component lies in the meromorphic stratum [0](5;-3,-2,-2) with a residue equal to 0 at the pole of order -3. Again <cit.> implies that there are 2 non isomorphic such differentials. As before the local degree of each family  is 2 and there are two distinct differential on a smooth curve which degenerate to the considered pointed curve.
 Now consider the curve in the third column of Table <ref>, when the pole of order -3 is on the top component. The differential on the top component in the stratum [0](1,0;-3). The differential on the bottom component is in the stratum [0](5;-3,-2,-2) with a residue equal to 0 at the marked pole of order -2. There is a unique such differential. Note that the prong numbers are equal to 1 and 2. Hence there are two such depending on which node we put the distinct prong numbers. There is only one class of prong-matching. Since locally at the nodes  the families are given by the equations x_1y_1 = t^2 and x_1y_1 = t, we deduce from Lemma <ref> that each family is a cover of order 3 to [1,1]. Hence there are 6 differentials on a smooth curve which arise from this case.
 Finally, we consider the fouth pointed curve where both poles are on the top component. The differential on this component is the locus [0](2,1;-2,-3). Again <cit.> gives that there is a unique such differential. The differential  on the bottom component is the unique differential in [0](5;-2,-3). Since there are two ways to assign the prong numbers at the nodes,  there are two on this pointed curve. Note that there is a unique class of prong-matching, hence there are two distinct multi-scale differentials on this pointed curve. Again each family is locally a cover of order 5 to [1,1]. Hence there are 10 differentials which come from this case.
Summing up all the contributions we conclude that there are 24 non isomorphic differentials in [1](5;-2,-3) on a smooth curve of genus 1. 



We now prove Theorem <ref> by  arguments similar to the ones used to prove Lemma <ref>. Recall that this theorem says that 
 the degree of π[1](6;-2,-2,-2) →[1,1] is 7. Note that by Theorem <ref>, the dimension of [1](6;-2,-2,-2) is equal to the dimension of [1,1], hence it make sense to compute the degree of the forgetful map. 


 According to Lemma <ref> there are 4 types of pointed curves on which there can exists a multi-scale differential in the closure of [1](6;-2,-2,-2). These curves are shown in the first row of Table <ref> (the entries of this table are explained after Table <ref>).
 
 Let us consider the pointed curves of the first column. The pull-back of the differential on the normalisation lies  in  the stratum [0](6;-2,-2,-2,-1,-1) with all the residues of the poles of order -2 equal to zero. There is a unique such differential, and it is not difficult to show that it leads to  a unique multi-scale differential on such pointed curve. Hence there is a unique differential on a smooth curve degenerating to this multi-scale differential.
 Consider the pointed curve in the second column. The differential on the top component is the unique one in the stratum [0](0,0;-2). The differential on the bottom component is in [0](6;-2,-2,-2,-2) with two poles having zero residue. There is a unique such by <cit.>. Moreover, since there is a unique class of prong-matching, there is a unique multi-scale differential on this pointed curve. The local equation of the family at the nodes is x_iy_i=t, so by Lemma <ref> the local degree of the family of pointed curves is 2. Since the family of differentials near the top component is given by ω_0 and the bottom by t ω_-1, the differentials on the isomorphic curves for parameters t and -t are distinct. Hence there are two differentials on a smooth genus 1 curves converging to this multi-scale differential.
 Consider the case of pointed curves given in the third column of Table <ref>. The differential on the top component is the unique differential in [0](1,1;-2,-2). The differential on the bottom component is in the stratum [0](6;-2,-3,-3) with two poles having zero residue. Proposition 2.3 of <cit.> implies that there is a unique such twisted differential. Since the prong-numbers are both equal to 2, there are 2 classes of prong-matchings. For each class the degree of the restriction of the stabilisation map to this family is 2. But since near the bottom differential the family looks like t^2ω_-1, the 2 differentials coincide (the marked poles are permuted). Hence there are 2 differentials on a smooth curve which degenerate to the multi-scale differentials in this case.
 In the last pointed curve of Table <ref>, the restriction ω_0 of the differential to the top component is the unique element in the locus [0](2,2;-2,-2,-2). Note that the top differential in [0](2,2;-2,-2,-2) has an automorphism group equal to /3. This can be done by checking directly that ω_0 = z^2(z^3-1)^2 dz or showing at the flat picture of it given in Figure <ref>. The differential on the bottom component is the unique element in [0](6;-4,-4). There is a unique such twisted differential but there are 3 classes of prong-matchings. Here we have a new phenomenon occurring. Indeed the automorphism group of ω_0 acts on the classes of prongs and hence on the multi-scale differentials. One class of prongs is represented in Figure <ref> and the acts of  the automorphism group consist of permuting cyclically the polar domains. Hence there is only one class prong-matching modulo isomorphism.  For the smoothing of this multi-scale differential the degree of the stabilisation map is 2. It is easy to see that the 2 differentials on isomorphic curves are indeed distinct. Hence there are 2  differentials on a smooth curve degenerating to this multi-scale differential.
 
 
 Summing up all the cases we obtain that there are 7 non isomorphic differentials in the locus [1](6;-2,-2,-2) on a general curve of genus 1.
 To conclude the proof, it suffices to show that the map π[1](6;-2,-2,-2) →[1,1] is unramified. To prove this, it suffices to show that there exit no twisted differentials of type (6;-2,-2,-2) of the second kind on a curve semi-stably equivalent to a smooth curve of genus 1. This follows from <cit.> by the fact that every twisted differential on such curve has a non-zero residue either at a node or at a smooth pole.


To conclude this section on enumerative problems related to differentials, we compute the degree of the map π[2](6;-2,-2)→[2], which is finite by Theorem <ref>.

The degree of the map π[2](6;-2,-2)→[2] is 644.

The proof of this proposition is by degeneration in the moduli space of multi-scale differentials. 


 Let us consider the stable curve X which is union of two general elliptic curves X_1 and X_2 attached to a point q. We denote by z the zero of order a, by w_1 and by w_2 the poles of order 2. We compute  the number of multi-scale differentials on X.
 There are six types of pointed curves (X;z,w_1,w_2) which can have a  multi-scale differential of type (6;-2,-2) of the second kind. The ones with z∈ X_1  are shown in Figure <ref> and the other stable curves are symmetric to these ones  with z∈ X_2.

 
In case (a), the differential ω_1 on X_1 is in the locus  [1](6;-2,-2,-2). Theorem <ref> gives that there are 7 non isomorphic such differentials. Moreover, the differential ω_2 on X_2 is the unique holomorphic differential on this elliptic curve. Since there are three ways of pasting these differentials together (one for each pole  of the  differentials ω_1), then there are d_a=21  such . Moreover since there is a unique class of prong-matchings there are 21 multi-scale differentials of this type.
In  case (b), the differential over X_1 is in the locus [1](6;-2,-4). Corollary <ref> gives that there are 
 6^2+4^2-10=42 such differentials.
  Moreover the differential on X_2 is one of the  3 differentials of [1](2;-2). Since there is a unique class of prong-matching, there exist d_b=42·  3=126 such multi-scale differentials.
In  case (c), the differential  on X_1 in  one of the 35 differentials of [1](6;-6).  The differential on X_2 is in [1](4;-2,-2). Hence Lemma <ref> implies that there are 5 such differentials. Finally there exist d_c=35 · 5=175 such multi-scale differentials.
Summing up all the contributions, there exist

    d = 2· (d_a + d_b +d_c)= 2·( 21 + 126 + 175) =644

multi-scale differentials in the closure of [2](6;-2,-2) on the curve X.
 To conclude that  the degree of π on X is equal to 644, it remains to show that there is a unique differential which converges to each multi-scale differential. Since the family of curve is given by xy = t at the node, there is a unique differential on nearby smooth curves which converges to each multi-scale differentials. 





















 §.§ The class of [3](6;-2) in  the Picard group of [3].




To compute the boundary classes of the divisor [3](6,-2) of the projection of [3](6,-2) in [3], we use the technique of test curves.  The test curves that we use are well-known and we follow the notation of <cit.> to denote them.



  
First curve.

The first curve A is obtained in the following way. Let X_2 be a curve of genus 2 and p a generic point of X_2. We glue at p a base point q of a pencil of plane cubic. 

 The intercession number of [3](6;-2) with A is equal to 0.






 It is sufficient to show that there is no twisted differential of type (6;-2) on a curve semi-stably equivalent to any curve of A.  Lemma <ref> tells us that it is enough to consider cases where the points z and w are on the smooth part of X. So we have four cases to consider, which we represent in Figure <ref>.
   
In case (a) the restriction ω_2 of ω to X_2 is in the stratum [2](4;-2). Since this stratum has a projective dimension equal to the dimension of [2], there is a finite number of points that can be the pole of ω_2. Hence no twisted differential of this form can exist, since the point p is generic.
In case (b), the same argument works since the differential  ω_2 is in [2](6;-4) whose projective dimension is equal to the one of [2].
In the case (c), the differential ω_2 has a zero of order 2 at p. So this point has to be a point. It is not possible because this point is generic on X_2.
In the case (d), the differential ω_2 is in the stratum [2](6;-2,-2). In addition, the global residue condition implies that all the residues are equal to 0. By Lemma <ref> the locus that parametrises these differentials is of dimension three. This implies that the point p has to be special on X_2, which gives the last impossibility.






  
Second curve.

The second curve C is obtained as follows. We choose a general curve X_2 of genus 2 and an elliptic curve X_1 which we paste together at points q_2∈ X_2 and q_1∈ X_1. The family C  is the family where q_2 varies over X_2.

 The intersection number between the divisor [3](6;-2) and the curve C is equal to 8792.



 A curve X in the family C is in the locus [3](6;-2) if and only if there is a multi-scale differential of type (6;-2) on a pointed curve semi-stably equivalent to X. Lemma <ref> implies that no marked point can be on a rational bridge between the two components X_1 and X_2. Moreover, the points cannot be together on a rational curve. Otherwise on this curve the differential has to be in the stratum [0](6;-2,-6). By <cit.> implies that the residues of the differential at the poles are not zero. Then the global residue condition cannot be fulfilled. Therefore, the possible curves belonging to C and in the divisor [3](6;-2) are of the types pictured in Figure <ref>.
 In case (a), the differential over X_1 is in the stratum [1](6;-6) and the differential on X_2 is in [2](4;-2). By Proposition <ref> there are d_2=2· 4^2· (-2)^2 -18 = 110 possible position for the points q_2. So there are 110 curves in the intersection of C and [3](6;-2) where the zero tends to X_1 and the pole to X_2. It remains to compute the multiplicity of these points. Note that on X_1, there are  d_1=6^2-1 = 35 possible non isomorphic differentials. Moreover the space of multi-scale differentials is smooth at these points. Hence this configuration contribute to d_a=d_1· d_2 = 3850 to the intersection number between C and [3](6;-2).
In case (b), the restriction of the multi-scale differential to X_1 is in [1](2;-2) and the restriction to X_2 is in [2](6;-4). So by Proposition <ref> there is d_2=2· 6^2· 4^2 -18 = 1134 possible positions for the point q_2. Hence there are 1134 curves of C having multi-scale differentials of this type. Moreover, since there are d_1=3 possible differentials on X_1 and the space is smooth at these points, each of this point contribute to 3 to the intersection number. So this case contribute to d_b=d_1· d_2 =3402 to the intersection number  between C and [3](6;-2).
In case (c), the restriction of the multi-scale differential to X_1 is in [1](6;-2,-4) and the restriction to X_2 is in [2](2). So there are 6 pointed curves that have a twisted differential of this type (one for each point). Moreover, Corollary <ref> tells us that there are 6^2 + 4^2 -10=42 differentials in [1](6;-2,-4) over X_1. So this case contribute to  d_b=6*42=252 to the intersection number.
In case (d), the restriction of the multi-scale differential to X_1 is in [1] and the restriction to X_2 is in [2](6;-2,-2). So Proposition <ref> implies that the number of curves in C  that lies in [3](6;-2) with a multi-scale differential of type (d) is 644. Since each of the intersection is simple and there is two choices for the pole of order 2, this case contribute to  1288 to the intersection number.
To conclude the proof, it is sufficient to add up the contributions of each type. Hence we have

    [3](6;-3) ∩[ C ]  = 3850 + 3402 + 252 + 1288 = 8792  .





  
Conclusion.

Now it remains to collect the information of this section to give the class αλ + βδ_0 + γδ_1 of [3](6;-2) in ([3])⊗.  The classical result of <cit.> implies that:  

    α +12 β - γ   =    0  ,
    
          -2γ   =    8792   .

Moreover we now by Equation (<ref>) that α=17108. 
Hence the class of [3](6;-2) in  
([3])⊗ is given by


    [ [3](6;-2) ] = 17108λ - 1792δ_0 - 4396 δ_1 .

Hence by <cit.> (see in particular Corollary 3.95) we have that the class of [3](6;-2) in the Picard group of the moduli space is given by

    [ [3](6;-2) ] = 17108λ- 3584δ_0 - 4396 δ_1 .

To conclude note that <cit.> gives that [3](6;-2) has one hyperelliptic components and two other components distinguished by parity.  We denote by [3]^ hyp(6;-2) the projection of the hyperelliptic component and by [3]^+(6;-2) and [3]^-(6;-2) the projection of the other components.
In the hyperelliptic case the zero and the pole are distinct points.  Since there are 8 such points, the class of  [3]^ hyp(6;-2) is  56 times the class of the hyperelliptic divisor in [3].  According to <cit.>, the class of the hyperelliptic locus is 9λ - δ_0 - 3δ_1. Hence  a direct consequence of Theorem <ref> is the following result.

 In the rational Picard group of the stack [3] we have
 
    [ [3]^+(6;-2) ] + [ [3]^-(6;-2) ] = 16604λ - 1736δ_0 - 3750 δ_1 .



 alpha
  

