DOUBLE HURWITZ NUMBERS AND THE INTERSECTION

FORM ON THE MODULI SPACE IN GENUS 0

M. Shapiro^{∗} and A. Vainshtein^{†}

∗ MPI and Department of Mathematics, Michigan State University East Lansing, MI 48824, USA, mshapiro@math.msu.edu

†MPI and Depts. of Mathematics and Computer Science, University of Haifa Mount Carmel, 31905 Haifa, Israel, alek@cs.haifa.ac.il

Abstract. We give an explicit formula for the double Hurwitz numbers in genus zero counting the number of coverings ℂP^{1} → ℂP^{1} with two branching points with a given branching behavior, as piecewise polynomials in the multiplicities of the preimages of the branching points.

§1. Introduction and results

Let μ = (μ_{1}, …, μ_{m}) be a partition of an integer d. The Hurwitz number h_{μ}^{g} is the number of genus g branched covers of ℂP^{1} with branching corresponding to μ over a fixed point (usually identified with ∞) and an appropriate number of fixed simple branching points. Recall that each cover is counted with the weight c^{−1}, where c is the number of automorphisms of the cover.

Hurwitz numbers possess a rich structure explored by many authors in different fields, such as algebraic geometry, representation theory, integrable systems, combinatorics, and mathematical physics. Being very far of trying to describe all these achievements, we mention only the so-called ELSV-formula (see [ELSV1, ELSV2, GrVa])

h_{μ}^{g} = \frac{(m + d + 2g − 2)!}{| Aut μ|} ∏_{i=i}^{m} \frac{μ_{i}^{μ_{i}}}{m_{i}!} ∫_{\overline{M}_{g,m}} \frac{1 − λ_{1} + λ_{2} ⋯ ± λ_{g}}{(1 − μ_{1}ψ_{1}) ⋯ (1 − μ_{m}ψ_{m})},

where Aut μ is the group of symmetries of the set μ, \overline{M}_{g,m} is the Deligne–Mumford compactification of the moduli space of genus g curves with m marked points, λ_{k} is a codimension k class, and ψ_{i} is a codimension 1 class (see the above mentioned papers for exact definitions).

Let ν = (ν_{1}, …, ν_{n}) be another partition of the same integer d. The double Hurwitz number h_{μ,ν}^{g} is the number of genus g branched covers of ℂP^{1} with branchings corresponding to μ and ν over two fixed points (in what follows we identify them with ∞ and 0, respectively) and an appropriate number of fixed simple branching points. We denote this number r_{μ,ν}^{g}; by the Riemann–Hurwitz theorem, r_{μ,ν}^{g} = m + n + 2g − 2. As before, each cover is counted with the weight c^{−1}, where c is the number of automorphisms of the cover. Evidently, h_{μ,ν}^{g} = h_{ν,μ}^{g}, so in what follows we assume, without loss of generality, that m ⩽ n.

Most of the known results concerning double Hurwitz numbers treat only the so-called one-part (or polynomial) case, when m = 1 and n is arbitrary. One-part double Hurwitz numbers in genus zero where studied in [ZL1] (see also [Z1] for an earlier version of the same result). It is proved there that

(1.1)  h_{(d),ν}^{0} = \frac{(n − 1)!}{| Aut ν|}d^{n−2};

recall that n − 1 = r_{(d),ν}^{0}. Another way to get the same result is suggested in [Sh1]. The aim of this paper is to get an ELSV-type formula for h_{(d),ν}^{g}. Two such formulas are derived; the first of them (see [Sh1, Sh3]) represents h_{(d),ν}^{g} as

h_{(d),ν}^{g} = \frac{r!}{| Aut ν|}d^{r−1} ∫_{Δ} ψ_{1}^{r−1},

where r = r_{(d),ν}^{g} = n + 2g − 1, Δ is a certain cycle in the moduli space \overline{M}_{g,n+1} and ψ_{1} has the same meaning as in the ELSV-formula. In the case g = 0 the cycle Δ coincides with the whole space \overline{M}_{0,n+1}, and since the integral is known to be equal to 1, we immediately get (1.1). The second formula (see [Sh1, Sh2]) represents h_{(d),ν}^{0} as

h_{(d),ν}^{0} = \frac{d^{n−2}}{| Aut ν|} ∫_{\overline{M}_{0,d(n−1)+2}} ψ_{1}^{n−2}Ψ_{2} ⋯ Ψ_{n+1}.

The definition of the classes Ψ_{i} is recursive and rather involved. The only example considered in [Sh1] is n = 2, in which case, after some effort, one gets the correct answer.

A different ELSV-type formula for the one-part double Hurwitz numbers is conjectured in [GJVa]. It is proved there for the cases g = 0, 1. The answers look as follows:

h_{(d),ν}^{0} = \frac{(n − 1)!}{| Aut ν|}d ∫_{\overline{M}_{0,n}} \frac{1}{(1 − ν_{1}ψ_{1}) ⋯ (1 − ν_{n}ψ_{n})},
h_{(d),ν}^{1} = \frac{(n + 1)!}{| Aut ν|}d ∫_{\overline{M}_{1,n+1}} \frac{1 − Λ_{2}}{(1 − ν_{1}ψ_{1}) ⋯ (1 − ν_{n}ψ_{n})},

where ψ_{i} are the same as in the ELSV formula, and Λ_{2} is a certain codimension 2 class pulled back from \overline{M}_{1,2}. For higher genera, the conjectured formula looks very similar, however, it is yet not clear over which moduli space should the integral be taken. On the computational side, [GJVa] suggests the following result, which is a common generalization of (1.1) and [SSV, Th. 6]:

h_{(d),ν}^{g} = \frac{r!}{| Aut ν|}d^{r−1}[t^{2g}] ∏_{k⩾1} (\frac{sinh kt/2}{kt/2})^{c_{k}} ,

where r = r_{(d),ν}^{g}, {c_{k}} is a finite sequence defined solely by ν, and [A]B is the coefficient of A in B. This result extend previous computations carried out in [Ku] for n = 1, 2.

Much less is known for general double Hurwitz numbers. In fact, there are two general results. First, it is proved in [Ok] that the exponent of the generating function for the double Hurwitz numbers is a τ-function for the Toda hierarchy of Ueno and Takasaki. Second, Theorem 2.1 in [GJVa] states that for fixed g, m, and n, double Hurwitz numbers are piecewise polynomial in variables μ_{1}, …, μ_{m}, ν_{1}, …, ν_{n}, and that the highest degree of this piecewise polynomial is constant and equal to m + n + 4g − 3. Moreover, it is proved in [GJVa] that for genus zero case this piecewise polynomial is homogeneous.

In a different direction, Theorem 4.1 of the same paper treats h_{μ,ν}^{g} as a function of g for fixed partitions μ and ν. In principle, any such function can be obtained by recursive computation; the only case done explicitly is m = n = 2 and μ_{1} > ν_{1} > ν_{2} > μ_{2}. Similar results are also obtained in [Ku] for all double Hurwitz numbers of degree at most 5.

Still another approach suggested in [GJVa] provides exact formulas for double Hurwitz numbers in genus zero when m = 2 or m = 3. In particular,

h_{(μ_{1},μ_{2}),ν}^{0} = \frac{n!}{d| Aut μ|} ∑ \frac{l(ρ)! ∏_{j⩾1} ρ_{j}}{| Aut ρ|| Aut σ|| Aut τ|}(μ_{1} − |σ|)(μ_{2} − |τ|)μ_{1}^{l(σ)−1}μ_{2}^{l(τ)−1},

where the summation is over partitions ρ = (ρ_{1}, …, ρ_{l(ρ)}), σ = (σ_{1}, …, σ_{l(σ)}), τ = (τ_{1}, …, τ_{l(τ)}) with ρ ∪ σ ∪ τ = ν and |σ| = ∑_{i=1}^{l(σ)} σ_{i} < μ_{1}, |τ| = ∑_{i=1}^{l(τ)} τ_{i} < μ_{2}.

Finally, the approach developed in [Z2] allows in principle to obtain formulas for h_{μ,ν}^{0} as functions of the degree d for a very special choice of the partitions μ and ν: μ = (α, 1, …, 1), ν = (β, 1, …, 1), α and β being fixed. For example, for α = β = 3 one gets

h_{3^{1}1^{d−3},3^{1}1^{d−3}}^{0} = \frac{3}{4}(27d^{2} − 137d + 180)\frac{d^{d−6}(2d − 6)!}{(d − 3)!}.

It is conjectured that for arbitrary α and β the number in question equals P_{αβ}(d)d^{d−α−β+1}(2d − α − β)!/d!, where P_{αβ} is a polynomial of degree α + β − 2 with rational coefficients; for example, P_{33}(d) = \frac{3}{4}(d − 1)(d − 2)(27d^{2} − 137d + 180).

In this note we give an explicit formula for the double Hurwitz numbers h_{μ,ν}^{0} as a sum of homogeneous piecewise polynomials of degree m + n − 3 over a set of weighted trees. To formulate the main result we need to introduce some notation. Let T = (V_{T} , E_{T}) be a tree with the vertex set V_{T} and the edge set E_{T}. A weight function is a map w: V_{T} ∪ E_{T} → ℤ_{⩾0} satisfying the following properties:

(1.2)

∑_{v∈V_{T}} w(v) = ∑_{e∈E_{T}} w(e) = W_{T} ,

w(v) ⩾ deg v − 3 for any v ∈ V_{T} ,

where deg v is the number of edges incident to v; observe that the second condition is nontrivial only for vertices of degree at least 4. The pair (T, w) is called a weighted tree, and W_{T} = ∑_{e∈E_{T}} w(e) is the total weight of the weighted tree. Two weighted trees (T_{1}, w_{1}) and (T_{2}, w_{2}) are called isomorphic if there exists a bijection V_{T_{1}} → V_{T_{2}} preserving incidence relations and weights. We denote by T (s) the set of all weighted trees satisfying condition |E_{T} | + W_{T} = s modulo this isomorphism. Slightly abusing notation, we will call the elements of T (s) weighted trees, and denote them (T, w).

Given a weighted tree (T, w) ∈ T (s) and a set S of size s + 3, define a labeling ℓ as a function ℓ: S → V_{T} satisfying condition

(1.3)  |ℓ^{−1}(v)| = w(v) − deg v + 3

for any v ∈ V_{T} (recall that the right hand side is always nonnegative due to (1.2)). Two labelings ℓ_{1}, ℓ_{2} of a weighted tree (T, w) are called equivalent if there exists an automorphism φ of (T, w) such that ℓ_{1} = φ ∘ ℓ_{2}. We denote by L(T, w) the set of all labelings of (T, w) modulo this equivalence. Abusing notation, we denote the elements of L(T, w) with the same letter ℓ and call them labelings. Let (T, w) be a weighted tree with a labeling ℓ. For any edge e ∈ E_{T} define the equivalence relation ∼_{e} on S by the following condition: s ∼_{e} t if the unique simple path between ℓ(s) and ℓ(t) in T does not pass through e. Evidently, this equivalence relation has exactly two equivalence classes corresponding to the two subtrees obtained from T after deleting e. Combined with condition (1.3), this means that each edge of T defines a partition of S into two parts of size at least 2 each.

Given two partitions μ and ν as above, define S = {1, …, n, ¯1, …, m̄}. The size of S is m + n, so it can be used for labeling trees in T (m + n − 3). Given such a labeling, define the defect δ_{ℓ}(e) of an edge e ∈ E_{T} by the following expression:

δ_{ℓ}(e) = ∑_{i∼_{e}¯1} ν_{i} − ∑_{ī∼_{e}¯1} μ_{i}.

Finally, for any e ∈ E_{T} put

ξ_{ℓ}(e) = \{ \begin{matrix}μ_{1} + (δ_{ℓ}(e))_{+}  if 1 ∼_{e} 2 ≁_{e} ¯1, \\ (δ_{ℓ}(e))_{+}  otherwise, \end{matrix}

where a_{+} = max{0, a}.

We are now ready to formulate our main result. Let x_{1}, x_{2}, … and y_{1}, y_{2}, … be two sets of formal variables corresponding to the edges e_{1}, e_{2}, … and vertices v_{1}, v_{2}, … numbered arbitrarily. For any edge e ∈ E_{T} with the endpoints v_{i} and v_{j} put y(e) = y_{i} + y_{j}. Similarly, for any vertex v ∈ V_{T} with the incident edges e_{i_{1}}, …, e_{i_{deg v}} put x(v) = x_{i_{1}} + ⋯ + x_{i_{deg v}}.

Theorem 1.1. The double Hurwitz numbers h_{μ,ν}^{0} with m ⩾ 1, n ⩾ 2 are given by (1.4)

h_{μ,ν}^{0} = \frac{r!(r − 1)!}{| Aut μ|| Aut ν|} ∑_{(T,w)∈T (r−1)}(−1)^{W_{T}} C(T, w) ∑_{ℓ∈L(T,w)} ∏_{e∈E_{T}} \frac{ξ_{ℓ}(e)^{w(e)+1}}{(w(e) + 1)!},

where

(1.5)  C(T, w) = [∏_{i=1}^{k} x_{i}^{w(e_{i})} ∏_{j=1}^{k+1} y_{j}^{w(v_{j})}] ∏_{v∈V_{T}} x(v)^{w(v)} ∏_{e∈E_{T}} y(e)^{w(e)},

r = r_{μ,ν}^{0} = m + n − 2, and [A]B denotes the coefficient of A in B.

Observe that the product ∏_{e∈E_{T}} ξ_{ℓ}(e)^{w(e)+1} is a homogeneous piecewise polynomial of degree r − 1.

The authors are grateful to Max-Planck-Institut für Matematik, Bonn, for hospitality in Summer 2004, and to Yu. Manin and S. Natanzon for useful discussions.

2. Computations

2.1 One-part double Hurwitz numbers. Consider first the case of one-part double Hurwitz numbers, when m = 1, and hence μ_{1} = d. It is easy to see that under this condition δ_{ℓ}(e) < 0 for any tree T, any edge e ∈ E_{T}, and any labeling ℓ, and therefore, ξ_{ℓ}(e) is either d or 0. Let us say that a labeling ℓ is nontrivial if it gives a nontrivial contribution to the right hand side of (1.4). It is easy to see that the contribution of each nontrivial labeling is proportional to d^{n−2}. Moreover, each nontrivial labeling possesses the following property: 1 ∼_{e} 2 and 1 ≁_{e} ¯1 for any edge e. The only tree that admits a labeling possessing this property is a path. Denote by v_{1} and v_{k+1} the endpoints of the path; all nontrivial labelings of the path are characterized by conditions

(2.1)  ℓ(1) = ℓ(2) = v_{1},  ℓ(¯1) = v_{k+1}.

Denote by P(s) the set of (equivalence classes of) all weighted paths (P, w) satisfying condition |E_{P} | + W_{P} = s, and by L^{∗}(P, w) the set of (equivalence classes of) labelings of (P, w) satisfying condition (2.1). It follows from Theorem 1.1 and the above discussion that

h_{(d),ν}^{0} = \frac{(n − 1)!}{| Aut ν|}d^{n−2}I_{n},

where

I_{n} = (n − 2)! ∑_{(P,w)∈P(n−2)}(−1)^{W_{P}} \frac{C(P, w)|L^{∗}(P, w)|}{∏_{e∈E_{T}} (w(e) + 1)!}.

Lemma 2.1. I_{n} = 1.

We were unable to find a combinatorial proof of Lemma 2.1. However, its algebrogeometric meaning is very transparent, and will be explained below in Section 3. As an immediate corollary we get expression (1.1) for the one-part double Hurwitz numbers.

2.2 Two-part double Hurwitz numbers: the extremal case. Let now m = 2, and assume in addition that μ_{1} ⩾ ν_{1} ⩾ ⋯ ⩾ ν_{n} ⩾ μ_{2}. Similarly to the previous case, δ_{ℓ}(e) is always negative. Therefore, the contribution of each nontrivial labeling is proportional to μ_{1}^{n−1}. The structure of graphs admitting nontrivial labelings, and the conditions characterizing nontrivial labelings are exactly the same as in the previous case. We therefore get

h_{(μ_{1},μ_{2}),ν}^{0} = \frac{n!}{| Aut μ|| Aut ν|}μ_{1}^{n−1}I_{n+1} for μ_{1} ⩾ ν_{1} ⩾ ⋯ ⩾ ν_{n} ⩾ μ_{2}.

Using again Lemma 2.1 we arrive to the following result.

Corollary 2.2. Assume that μ_{1} ⩾ ν_{1} ⩾ ⋯ ⩾ ν_{n} ⩾ μ_{2}, then the two-part double Hurwitz numbers h_{(μ_{1},μ_{2}),ν}^{0} are given by

h_{(μ_{1},μ_{2}),ν}^{0} = \frac{n!}{| Aut μ|| Aut ν|}μ_{1}^{n−1}.

Observe that the only situation when the factor | Aut μ| in the denominator is nontrivial arises when n = 2 and μ_{1} = μ_{2} = ν_{1} = ν_{2}.

2.3 Five ramification points: complete description. The first case not covered by the above two formulas arises when m + n = 5. Since m ⩽ n, and the case m = 1 is covered by (1.1), we remain with the case m = 2 and μ_{1} ⩾ μ_{2}, ν_{1} ⩾ ν_{2} ⩾ ν_{3}. Below we derive from Theorem 1.1 an explicit expression for double Hurwitz numbers in this case.

Since r = m + n − 2 = 3, the summation in (1.4) goes over T (2). This set contains exactly two weighted trees shown on Fig. 1; the numbers above the trees are weights of edges, those below the trees are weights of vertices.

Fig. 1. The set T (2): a) (T_{1}, w_{1}); b) (T_{2}, w_{2})

For the first tree one has C(T_{1}, w_{1}) = [x_{1}y_{1}](y_{1} + y_{2})x_{1} = 1, for the second one, C(T_{2}, w_{2}) = [1]1 = 1. To calculate the contributions of labelings, define δ(a, b), a, b ∈ {1, 2, 3, ¯1, ¯2}, by the following rules:

δ(i, j) = 0,  i, j = 1, 2, 3,
δ(ī, j̄) = 0,  i, j = 1, 2,
δ(i, j̄) = (−1)^{j}(μ_{j} − ν_{i}),  i = 1, 2, 3, j = 1, 2.

Consider an arbitrary edge e of T_{1} or T_{2}. According to (1.3), it is incident to a unique vertex having exactly two preimages under each labeling. Let ℓ be a labeling, a, b ∈ {1, 2, 3, ¯1, ¯2} be the two preimages. It is easy to check that δ_{ℓ}(e) = δ(a, b). Besides, 1 ∼_{e} 2 ≁_{e} ¯1 if and only if {a, b} = {1, 2}, {¯1, ¯2}, {3, ¯1}. Therefore, there are at most 8 nontrivial labelings of (T_{1}, w_{1}), and at most 8 nontrivial labelings of (T_{2}, w_{2}). Adding all the contributions, we get

h_{(μ_{1},μ_{2}),(ν_{1},ν_{2},ν_{3})}^{0} = \frac{6}{| Aut μ|| Aut ν|}P (μ_{1}, μ_{2}, ν_{1}, ν_{2}, ν_{3}),

where

P (α, β, γ, δ, ε) = α^{2}+2α((β−γ)_{+}+(β−δ)_{+}+(β−ε)_{+})+2(γ−α)_{+}((β−δ)_{+}+(β−ε)_{+})

+ 2(δ − α)_{+}(β − ε)_{+} − (β − γ)_{+}^{2} − (β − δ)_{+}^{2} − (β − ε)_{+}^{2} − (γ − α)_{+}^{2} − (δ − α)_{+}^{2};

while deriving this formula it was taken into account that μ_{1} + μ_{2} = ν_{1} + ν_{2} + ν_{3} implies (ν_{3} − μ_{1})_{+} = 0 and (ν_{2} − μ_{1})_{+}(μ_{2} − ν_{1})_{+} = 0.

3. Proofs

Denote by p_{i}, i = 1, …, n, the ramification points with multiplicities ν_{i}, and by pj̄, j = 1, …, m, the ramification points with multiplicities μ_{j}. Define the Hurwitz space H_{μ,ν} as the space of meromorphic functions on genus 0 curves with zeros at p_{i} of multiplicities ν_{i} and poles at pī of multiplicities μ_{i}. More exactly, denote by U = \overline{M}_{0,m+n+1} the universal curve over \overline{M}_{0,m+n}. Consider the relative sheaf O(∑ μ_{i}pī − ∑ ν_{i}p_{i}), i.e., the sheaf of local sections of meromorphic functions on the fibers of U with poles at pī of multiplicities μ_{i} and zeroes at p_{i} of multiplicities ν_{i}. By H_{μ,ν} we denote its push-forward to \overline{M}_{0,m+n}.

It is easy to see that H_{μ,ν} is a vector bundle over \overline{M}_{0,m+n}. The following result is an analog of Lemma 5.3 from [ELSV2].

Lemma 3.1.

h_{μ,ν}^{0} = \frac{(m + n − 2)!}{| Aut μ|| Aut ν|} ∫_{\overline{M}_{0,m+n}} s_{top}(H_{μ,ν}),

where s_{top}(H_{μ,ν}) is the top Segre class of H_{μ,ν}.

The proof is similar to the proof of Lemma 5.3 from [ELSV2] (see also [LZ, Thm. 4] for a similar result).

Since n ⩾ 2, for a smooth curve we can choose a local coordinate x such that x(p_{1}) = 0, x(p_{2}) = 1, x(p¯1) = ∞. Therefore, any rational function with the prescribed ramifications can be written as

(3.1)  f(x) = C \frac{x^{ν_{1}}(x − 1)^{ν_{2}}(x − a_{3})^{ν_{3}} … (x − a_{n})^{ν_{n}}}{(x − a¯2)^{μ_{2}} … (x − am̄)^{μ_{m}}} ,

where a_{j} = x(p_{j}), aj̄ = x(pj̄). We see that H_{μ,ν} is a line bundle, and the constant C is a coordinate in its fiber. It is important to note that, since the degree of functions in H_{μ,ν} is a constant, vanishing of an element of H_{μ,ν} does not mean that the corresponding map from a curve Γ to ℂP^{1} becomes the constant map to point 0. Rather, it means that the coefficient C in (3.1) vanishes, and the corresponding function degenerates to a prestable map from a prestable curve Γ̃ to ℂP^{1}, where Γ̃ = Γ∪⋃ī Γī. A new irreducible component Γī emerges from each marked point pī. The restriction of the function on Γī has a form z^{μ_{i}}, where z is a local coordinate on Γī vanishing at the double point. The same holds true when functions vanish on irreducible components of singular curves.

Recall that the total Segre class of a line bundle is the inverse of its total Chern class. Since H_{μ,ν} is one-dimensional, we have c(H_{μ,ν}) = 1 + c_{1}(H_{μ,ν}), and hence the formula in Lemma 3.1 translates to

h_{μ,ν}^{0} = \frac{(m + n − 2)!}{| Aut μ|| Aut ν|} ∫_{\overline{M}_{0,m+n}}(−1)^{m+n−3}c_{1}^{m+n−3}(H_{μ,ν}).

To calculate the first Chern class of H_{μ,ν}, consider the section σ = {C = 1}. It is evident that zeroes and poles of σ are located on the boundary divisors of \overline{M}_{0,m+n}. Given a subset K ⊂ \{1, …, n, ¯1, …, m̄\} such that 2 ⩽ |K| ⩽ m + n − 2, denote by D_{K} the boundary divisor containing two-component curves with the points p_{i}, i ∈ K, and pj̄, j̄ ∈ K, lying on one of the components, and all the other marked point lying on the other component. Besides, denote ν_{K} = ∑_{i∈K} ν_{i} and μ_{K} = ∑_{j̄∈K} μ_{j}. Taking into account that D_{K} = D_{K^{c}}, where K^{c} is the complement of K, we may assume that ¯1 ∈ K and relate each divisor to one of the following two groups:

(i) K ∩ {1, 2} ≠ ∅;

(ii) K ∩ {1, 2} = ∅.

The poles and zeros of the section σ are described by the following proposition.

Lemma 3.2. (a) σ does not vanish on any of D_{K}.

(b) σ has a pole of order (ν_{K} − μ_{K})_{+} on any D_{K} related to group (i).

(c) σ has a pole of order μ_{1} + (ν_{K} − μ_{K})_{+} on any D_{K} related to group (ii).

Proof. To understand the behavior of σ in neighborhood of boundary components we consider the degeneration of σ along a 1-parameter family of curves. We denote the parameter by ε and assume that curves in the family are nonsingular for nonzero values of ε. For ε = 0 a curve has at most two irreducible components.

Consider case (i) above. Our 1-parameter family is given by x(p_{i}(ε)), x(pj̄(ε)) → a_{0}, |a_{0}| < ∞, as ε → 0 for i, j̄ ∈ K^{c}. We can assume x(p_{i}) − a_{0} = εA_{i}, x(pj̄) − a_{0} = εAj̄ for some finite A_{i}, Aj̄. The curve Γ(ε) splits into two components Γ_{1} ∪ Γ_{2} as ε → 0, such that x(Γ_{1} ∩ Γ_{2}) = a_{0}, and p¯1 ∈ Γ_{1} whereas Γ_{2} contains p_{i}, pj̄, i, j̄ ∈ K^{c}. Consider first the case ν_{K} − μ_{K} > 0. Since the defect is positive for Γ_{1}, the nontrivial part is the restriction of σ on Γ_{2}. Choose a coordinate y on Γ_{2} using x − a_{0} = εy. Then

σ → \frac{∏_{i∈K^{c}}(ε(y − A_{i}))^{ν_{i}} ∏_{i∈K}(a_{0} − a_{i})^{ν_{i}}}{∏_{j̄∈K^{c}}(ε(y − Aj̄))^{μ_{j}} ∏_{j̄∈K\\{¯1\}}(a_{0} − aj̄)^{μ_{j}}}

= \frac{1}{ε^{ν_{K}−μ_{K}}} \frac{∏_{i∈K^{c}}(y − A_{i})^{ν_{i}} ∏_{i∈K}(a_{0} − a_{i})^{ν_{i}}}{∏_{j̄∈K^{c}}(y − Aj̄)^{μ_{j}} ∏_{j̄∈K\\{¯1\}}(a_{0} − aj̄)^{μ_{j}}} .

Hence, σ has a pole of order ν_{K} − μ_{K} on D_{K}.

Consider now the case ν_{K} − μ_{K} < 0. Since the defect is positive for Γ_{2}, the nontrivial part is the restriction of σ on Γ_{1}. On Γ_{1} we have

σ → \frac{∏_{i∈K^{c}}(x − a_{0})^{ν_{i}} ∏_{i∈K}(x − a_{i})^{ν_{i}}}{∏_{j̄∈K^{c}}(x − a_{0})^{μ_{j}} ∏_{j̄∈K\\{¯1\}}(x − aj̄)^{μ_{j}}} = (x − a_{0})^{ν_{K}−μ_{K}} \frac{∏_{i∈K}(x − a_{i})^{ν_{i}}}{∏_{j̄∈K\\{¯1\}}(x − aj̄)^{μ_{j}}} .

Hence, σ has neither poles nor zeros on D_{K}. The same is true also in the case ν_{K} − μ_{K} = 0.

Consider now case (ii) above. Our 1-parameter family is given by the condition x(p_{i}(ε)), x(pj̄(ε)) → ∞ as ε → 0 for i, j̄ ∈ K. We can assume x(p_{i}) = A_{i}/ε, x(pj̄) = Aj̄/ε for some finite A_{i}, Aj̄. Consider first the case ν_{K} − μ_{K} > 0. Then the nontrivial part of σ is the restriction on Γ_{1}, where we have

σ → \frac{∏_{i∈K}(x − A_{i}/ε)^{ν_{i}} ∏_{i∈K^{c}}(x − a_{i})^{ν_{i}}}{∏_{j̄∈K\\{¯1\}}(x − Aj̄/ε)^{μ_{j}} ∏_{j̄∈K^{c}}(x − aj̄)^{μ_{j}}}

= \frac{1}{ε^{ν_{K}−μ_{K}+μ_{1}}} \frac{∏_{i∈K}(−A_{i}) ∏_{i∈K^{c}}(x − a_{i})^{ν_{i}}}{∏_{j̄∈K\\{¯1\}}(−Aj̄) ∏_{j̄∈K^{c}}(x − aj̄)^{μ_{j}}} .

Hence, σ has a pole of order ν_{K} − μ_{K} + μ_{1} at D_{K}. Finally, it is easy to see that for ν_{K} − μ_{K} ⩽ 0, σ has a pole of order μ_{1} at D_{K}. □

Slightly abusing notation, we use the same symbol D_{K} to denote the dual cohomology class of the boundary divisor D_{K}. Therefore, by Lemma 3.2,

c_{1}(H_{μ,ν})

= − (  ∑_{\begin{matrix}K∩\{1,2\}≠∅ \\ 2⩽|K|⩽m+n−2 \end{matrix}}(ν_{K} − μ_{K})_{+}D_{K} + ∑_{\begin{matrix}K∩\{1,2\}=∅ \\ 2⩽|K|⩽m+n−2 \end{matrix}} (μ_{1} + (ν_{K} − μ_{K})_{+}) D_{K} ) .

To complete the computation, it is convenient to introduce some additional notation. Let S be a finite set. A 2-partition of S is a pair (S_{1}, S_{2}) such that S_{1}, S_{2} ≠ ∅ and S_{1} = S_{2}^{c}. Two 2-partitions (S_{1}, S_{2}) and (S′_{1}, S′_{2}) are called non-crossing if there exist S_{i} and S′_{j} such that S_{i} ∩ S′_{j} = ∅. A family C of 2-partitions is called crossing free if any two 2-partitions in C are non-crossing.

Let T = (V_{T} , E_{T}) be an arbitrary tree, and ℓ: S → V_{T} be an arbitrary mapping. Each edge e ∈ E_{T} defines two subtrees obtained from T by deleting e. The full preimages of the vertex sets of these subtrees under ℓ induce a 2-partition of S; in other words s, t ∈ S belong to the same part if and only if ℓ(s) and ℓ(t) belong to the same subtree. It is easy to see that the family of such 2-partitions of S corresponding to all edges in E_{T} is crossing free; we denote it by C(T). By the Edmonds–Giles theorem [EdGi, Thm. 8.3], a family C is crossing free if and only if there exists a tree T and a mapping ℓ : S → V_{T} such that C = C(T). Moreover, this tree is unique when the number of its edges is minimal, that is, different edges define different 2-partitions of S. The mapping ℓ in this case is unique as well.

Let M = D_{K_{1}}^{γ_{1}} ⋯ D_{K_{k}}^{γ_{k}} be an arbitrary monomial of total degree m + n − 3 in variables D_{K} ∈ H^{2}(\overline{M}_{0,m+n}). We assign to each D_{K} the 2-partition (K, K^{c}) of the set S = {1, …, n, ¯1, …, m̄}, and to the monomial M, the family C_{M} of 2-partitions corresponding to D_{K_{1}}, …, D_{K_{k}}. According to [Ke, Thm. 1], ∫_{\overline{M}_{0,m+n}} M can be non-zero only if C_{M} is crossing free.

Assume that C_{M} is crossing free, and let T_{M}, ℓ_{M} be the unique minimal tree and the mapping S → V_{T_{M}} corresponding to C_{M} by the Edmonds–Giles theorem. The geometric meaning of T_{M} and ℓ_{M} is transparent. Consider the (k + 1)-component stable curve corresponding to the set {D_{K_{1}}, …, D_{K_{k}}}. The vertices of T_{M} are the components of this curve, the edges of T_{M} correspond to the double points, and the full preimage of a vertex under ℓ_{M} is the set of marked points lying on the corresponding component.

Denote by e_{i} the edge of T_{M} corresponding to D_{K_{i}}. Let us define a weight function w_{M}: V_{T_{M}} ∪ E_{T_{M}} → ℤ_{⩾0}. For an edge e_{i} ∈ E_{T_{M}} put w_{M}(e_{i}) = γ_{i} − 1. For a vertex v ∈ V_{T_{M}} put w(v) = |ℓ_{M}^{−1}(v)| + deg v − 3. It is easy to see that the weight function w_{M} thus defined satisfies both conditions in (1.2). The nonnegativity of w_{M}(v) is evident for vertices with degree at least 3. For vertices of degree 2 it follows from the minimality of the tree, and for vertices of degree 1, from the stability condition 2 ⩽ |K| ⩽ m + n − 2. Moreover, the total weight of the tree equals m + n − 3 − |E_{T_{M}} |, hence (T_{M}, w_{M}) ∈ T (m + n − 3). Finally, condition (1.3) is satisfied, and hence ℓ_{M} is a labeling of (T_{M}, w_{M}).

Lemma 3.3. Let M = D_{K_{1}}^{γ_{1}} ⋯ D_{K_{k}}^{γ_{k}} be a monomial of total degree m + n − 3 such that C(M) is crossing free, and let (T_{M}, w_{M}) ∈ T (m + n − 3) be the corresponding weighted tree. Then

∫_{\overline{M}_{0,m+n}} M = (−1)^{W_{T_{M}}} C(T_{M}, w_{M})

with C(T, w) defined by (1.5).

Proof. Following [Ka], consider the linear system in variables f_{v,e}, v ∈ V_{T_{M}}, e ∈ E_{T_{M}}, e ∈ st(v):

f(v_{1}, e) + f(v_{2}, e) = w_{M}(e), e = (v_{1}, v_{2}),  ∑_{e∈st(v)} f(v, e) = w_{M}(v),

where st(v) is the set of edges incident to v. It is easy to see that this system, though overdefined, has a unique solution. The main theorem of [Ka] claims that

∫_{\overline{M}_{0,m+n}} M = ∏_{v∈V_{T_{M}}} (−1)^{w_{M} (v)} \frac{w_{M}(v)!}{∏_{e∈st(v)} f(v, e)!^{2}} ∏_{e∈E_{T_{M}}} w_{M}(e)!

provided all f(v, e) are nonnegative, and ∫_{\overline{M}_{0,m+n}} M = 0 otherwise. This formula can be rewritten as

∫_{\overline{M}_{0,m+n}} M

= (−1)^{W_{T_{M}}} ∏_{\begin{matrix}v∈V_{T_{M}} \\ e_{i}^{v}∈st(v) \end{matrix}} \binom{w_{M}(v)}{f(v, e_{1}^{v}) ⋯ f(v, e_{deg v}^{v})} ∏_{\begin{matrix}e∈E_{T_{M}} \\ e=(v_{1}^{e},v_{2}^{e}) \end{matrix}} \binom{w_{M}(e)}{f(v_{1}^{e}, e) f(v_{2}^{e}, e)}.

Observe that according to the definition of a multinomial coefficient, \binom{a}{b_{1}···b_{t}} = 0 if b_{i} < 0 for some i, 1 ⩽ i ⩽ t. Therefore, the above formula is correct independently of the signs of f(v, e)’s. It remains to notice that the double product in the right hand side is exactly the coefficient C(T_{M}, w_{M}) defined by (1.5). □

Theorem 1.1 follows easily from the above discussion.

It remains to prove Lemma 2.1. Denote by ψ¯1 the first Chern class of the line bundle over \overline{M}_{0,m+n} whose fiber is the cotangent space to the curve at p¯1. It is well known that

(3.1)  ∫_{\overline{M}_{0,m+n}} ψm+n−3 ; ¯1 = 1.

We are going to prove that the above integral equals I_{m+n−3}. Indeed, it is easy to see that ψ¯1 = ∑ D_{K}, where the sum is taken over all K ⊂ {1, …, n, ¯1, …, m̄} such that 1, 2 ∉ K, ¯1 ∈ K, 2 ⩽ |K| ⩽ m + n − 2. Let M be a monomial in variables D_{K} of total degree m + n − 3 with K satisfying the above conditions. It is clear from above that if M has a nontrivial contribution to the integral (3.1) then C(M) is crossing free. Moreover, the corresponding tree T_{M} is a path with the endpoints ℓ_{M}(1) = ℓ_{M}(2) and ℓ_{M}(¯1). Indeed, if this is not the case then there exists an edge of T_{M} such that 1, 2 and ¯1 are mapped by ℓ_{M} to the same of two subtrees defined by e, in a contradiction with the conditions defining possible subsets K. Observe that the above conditions on ℓ_{M} coincide with conditions (2.1) on the labeling of the path. Thus to get Lemma 2.1 it remains to apply Lemma 3.3.

References

[EdGi] J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Ann. od Discrete Math. 1 (1977), 185–204.

[ELSV1] T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein, On Hurwitz numbers and Hodge integrals, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 1175-1180.

[ELSV2] T. Ekedahl, S. Lando, M. Shapiro,and A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001), 297–327.

[GJVa] I. Goulden, D. Jackson, and R. Vakil, Towards the geometry of double Hurwitz numbers (2003), preprint math.AG/0309440.

[GrVa] T. Graber and R. Vakil, Hodge integrals and Hurwitz numbers via virtual localization, Compositio Math. 135 (2003), 25–36.

[Ka] R. Kaufmann, The intersection form in H^{∗}(\overline{M}_{0n}) and the explicit Künneth formula in quantum cohomology, Internat. Math. Res. Notices (1996), no. 19, 929–952.

[Ke] S. Keel, Intersection theory of moduli space of stable N-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545–574.

[Ku] S. Kuleshov, Ramified coverings of S^{2} with two degenerate branching points (2003), preprint MPIM2001-62.

[LZ] S. Lando and D. Zvonkine, Counting ramified coverings and intersection theory on Hurwitz spaces I (Cohomolgy of Hurwitz spaces) (2003), preprint math.AG/0303218.

[Ok] A. Okounkov, Toda equations for Hurwitz numbers, Math. Res. Lett. (2000), no. 7, 447–453.

[Sh1] S. Shadrin, Geometry of meromorphic functions and intersections on moduli spaces of curves, Int. Math. Res. Not. (2003), no. 38, 2051–2094.

[Sh2] S. Shadrin, Hurwitz numbers of generalized polynomials, and cycles of two-point branchings, Uspekhi Mat. Nauk 58 (2003), 197–198.

[Sh3] S. Shadrin, Polynomial Hurwitz numbers and intersections on \overline{M}_{0,k}, Funct. Anal. Appl. 37 (2003), 78–80.

[SSV] B. Shapiro, M. Shapiro, and A. Vainshtein, Ramified coverings of S^{2} with one degenerate branching point and enumeration of edge-oriented graphs, Adv. in Math. Sci. 34 (1997), 219–228.

[Z1] D. Zvonkine, Multiplicities of the Lyashko–Looijenga map on its strata, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 1349–1353.

[Z2] D. Zvonkine, Counting ramified coverings and intersection theory on Hurwitz spaces II (Local structure of Hurwitz spaces and combinatorial results) (2003), preprint math.AG/0304251.

[ZL1] D. Zvonkin and S. Lando, On multiplicities of the Lyashko–Looijenga mapping on strata of the discriminant, Funct. Anal. Appl. 33 (1999), 178–188.