Counting ramified coverings and intersection theory on spaces of rational functions I

(Cohomology of Hurwitz spaces)

S. K. Lando^{∗}, D. Zvonkine^{†}

May 13, 2003

To Sasha Mednykh on the occasion of his 50th birthday

Abstract

The Hurwitz space is a compactification of the space of rational functions of a given degree. The Lyashko–Looijenga map assigns to a rational function the set of its critical values. It is known that the number of ramified coverings of ℂP^{1} by ℂP^{1} with prescribed ramification points and ramification types is related to the degree of the Lyashko–Looijenga map on various strata of the Hurwitz space. Here we explain how the degree of the Lyashko–Looijenga map is related to the intersection theory on this space. We describe the cohomology algebra of the Hurwitz space and prove several relations between the homology classes represented by various strata.

1 Introduction

1.1 Statement of the problem

In a series of two papers, the present one and the second one by the second author [15], we continue the study of the Hurwitz problem [7] concerning counting ramified coverings of the 2-sphere. Roughly speaking, this problem can be formulated as follows:

∗Russian–French Laboratory for Mathematics, Informatics, and Mathematical Physics of the Independent University of Moscow and Institute for System Research, RAS; partly supported by the Russian Foundation for Basic Research (project 01-01-00660), NWO-RFBR (project 047.008.005), and INTAS (project 00259), e-mail: lando@lando.mccme.rssi.ru

†bât. 425, Université Paris-Sud, 91400 Orsay, France. e-mail: dimitri.zvonkine@math.u-psud.fr Both authors are partially supported by the RFBR grant 02-01-22004.

Given a set of fixed ramification points on the target 2-sphere and a set of ramification types over these points, count the number of non-isomorphic ramified coverings S → S^{2} by a 2-surface S, having the prescribed ramification types over the prescribed ramification points.

Two coverings are considered to be isomorphic if there is a homeomorphism of the covering surfaces taking the first covering to the second one. In fact, we count isomorphism classes with the weight equal to the inverse order of the group of automorphisms of the class; in this form the problem is more natural, and admits a lot of applications.

In a more modern setting this problem can be reformulated as the computation of the Gromov–Witten invariants of the complex projective line. A detailed exposition of the history of the problem and a description of various approaches to its solution can be found in [12].

Hurwitz [8] himself gave an explicit answer to the problem in the case, where the covering surface is also a sphere, and there is one ramification point with arbitrary ramification, while all others are points of simple ramification. In spite of the importance of the problem, only recently Hurwitz’s original results were extended to more general cases and treatable answers were obtained:

• I. Goulden and D. Jackson [5] solved the combinatorial problem that allows one to give the answer to Hurwitz’s problem for polynomials (that is, ramified coverings of the sphere by a sphere having a point with a single preimage and arbitrary ramification over other points); later their results were reestablished in [13] and [14] by absolutely different methods;

• T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein [2] gave an expression for the number of ramified coverings of the sphere by a surface of arbitrary genus with a single point of arbitrary ramification type and all other points of simple ramification, in terms of intersection indices on moduli spaces of curves (when the covering surface is the sphere, the answer given by the formula coincides with Hurwitz’s one); another proof of the formula was given in [6].

Here we address the special case of the Hurwitz problem, where the covering surface is also a sphere, but in contrast to the Hurwitz case, ramification points of arbitrary type are allowed. We are far from solving the problem in this generality, but we suggest a general approach, and show how it works in some special cases, thus producing new enumerative results.

To a ramification point (= a critical value) of multiplicity d we assign a partition κ of d, called the ramification type. The elements of κ are the multiplicities of the critical points that correspond to our critical value.

Recall the Hurwitz theorem.

Theorem 1 (Hurwitz) The number h_{κ} of degree n coverings of the sphere by the sphere, having the ramification type κ = (k_{1}, …, k_{m}) over one point, d = k_{1}+…+k_{m}, while all other ramification points are fixed and simple, is

h_{κ} = \frac{(2n − 2 − d)!}{(n − d − m)! |Aut(κ)|} ∏_{i=1}^{m} \frac{(k_{i} + 1)^{k_{i}+1}}{(k_{i} + 1)!} n^{n−d−3};

here |Aut(κ)| is the order of the automorphism group of the partition κ, 2n − 2 − d is the number of nondegenerate ramification points, and n − d − m is the number of simple preimages of the multiple ramification point.

As a consequence of our approach we obtain, in the second part of this paper [15], for example, the following result.

Theorem 2 The number h_{2,2} of degree n coverings of the sphere by the sphere, having two double critical points and 2n−6 simple critical points, all the 2n−4 critical values being fixed, is

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

As far as we know, this formula, as well as similar formulas for some other ramification types, is new.

1.2 The basis of the approach

First of all we reduce the problem to the calculation of the degree of some map, called the Lyashko–Looijenga map. This step is now standard. The Lyashko–Looijenga map (below, the LL map) takes a meromorphic function to the set of its critical values. Its source space can be chosen in a variety of ways; here we define the LL map on the Hurwitz space H_{n} constructed in [2]. It is the space of stable maps from genus zero complex curves to the projective line, having trivial ramification over infinity. All spaces of functions possessing degenerate ramification are considered as subvarieties in this space. These subvarieties form a stratification of the Hurwitz space.

Theorem 3 The number h_{{κ_{1},...,κ_{c}}} of ramified coverings of the sphere, having the ramification types κ_{1}, …, κ_{c} over prescribed ramification points with multiplicities d_{1}, …, d_{c}, is given by the formula

h_{\{κ_{1},...,κ_{c}\}} = \frac{1}{n!} \frac{|Aut\{κ_{1}, …, κ_{c}\}|}{|Aut\{d_{1}, …, d_{c}\}|} μ_{\{κ_{1},...,κ_{c}\}}

where μ_{{κ_{1},...,κ_{c}}} is the degree of the LL map restricted to the stratum Σ_{{κ_{1},...,κ_{c}}} consisting of functions with these ramification types.

This is an instance of a general situation, see e.g. [12]. Indeed, if we fix a ramified covering f : S → S^{2} and choose a complex structure on the target sphere S^{2}, then there exists a unique complex structure on the covering surface S making the function f into a meromorphic function. This complex structure is produced by the Riemann construction. Hence, there is a one-to-one correspondence between ramified coverings with fixed ramification points of given types and meromorphic functions with fixed critical values of the same types. The latter number is exactly the degree of the LL map on the corresponding moduli space. The coefficient 1/n! results from the fact that in our construction of the space H_{n} we choose a numbering of the n poles of each rational function. The factor

\frac{|Aut\{κ_{1}, …, κ_{c}\}|}{|Aut\{d_{1}, …, d_{c}\}|}

is due to the fact that if we permute the ramification types κ_{1}, …, κ_{c} preserving the multiplicities d_{1}, …, d_{c}, we obtain a new set of ramified coverings that lie on the same stratum Σ_{{κ_{1},...,κ_{c}}} and have the same image under the LL map.

The space H_{n} is, in fact, a vector bundle over the Deligne–Mumford moduli space \overline{M}_{0;n} of stable genus zero curves. The latter space is a smooth projective variety. The multiplicative group ℂ^{∗} of nonzero complex numbers acts on this bundle fiberwise. This action consists in just multiplying a meromorphic function by a constant. It preserves the stratification of the space H_{n} because the multiplication does not change the type of singularities. Deleting the zero section of the bundle and taking the quotient modulo this action we reduce the variety to the projectivization ℙH_{n} of the vector bundle H_{n}.

The projectivization ℙH_{n} carries the tautological line bundle T → ℙH_{n}, and a natural cohomology class Ψ_{n} ∈ H^{2}(ℙH_{n}), Ψ_{n} = c_{1}(T^{∨}), the class of a hyperplane section. The degree of the Lyashko–Looijenga map restricted to a subvariety in H_{n} is related to the intersection index of the subvariety with the complementary power of the class Ψ_{n}. Namely, the following statement is true.

Theorem 4 The degree μ_{{κ_{1},...,κ_{c}}} of the LL map restricted to the stratum Σ_{{κ_{1},...,κ_{c}}} is equal to

|Aut{d_{1}, …, d_{c}}| ⟨ℙΣ_{{κ_{1},...,κ_{c}}}, Ψ_{n}^{d}⟩.

Here d is the dimension of the stratum ℙΣ_{{κ_{1},...,κ_{c}}}, d_{i} is the multiplicity of the ith ramification point (or the sum of the elements of κ_{i}), and ⟨·, ·⟩ is the coupling between a homology and a cohomology class.

This theorem is proved in Section 3.2.

1.3 Cohomology of the Hurwitz spaces and Kazarian’s principle

Theorem 4 implies that our approach requires the study of the cohomology ring H^{∗}(ℙH_{n}). All cohomology groups we consider are with complex coefficients, and we do not specify the coefficients explicitly. We need

• a reasonable description of this ring, say, in terms of generators and relations;

• reasonable expressions for the cohomology classes of the strata and the class Ψ_{n} in terms of the generators.

The solution of the first problem is known since the space H_{n} is a vector bundle over the moduli space of stable rational curves, whose cohomology is known. The second problem seems to be much more difficult.

In fact, we need less than the whole cohomology ring. The symmetric group S_{n} acts on the space H_{n} by permuting the indices, whence it acts on the cohomology space H^{∗}(ℙH_{n}). All the cohomology classes we are interested in are symmetric with respect to this action. Therefore, we only need to know the S_{n}-symmetric part of the cohomology algebra H^{∗}(ℙH_{n}). This is an additional problem, but its solution can lead to a simplification of the decompositions of the strata.

The situation would become even easier if we restrict ourselves to the subalgebra in the cohomology algebra H^{∗}(ℙH_{n}) generated by the classes of the strata.

Consider two closed codimension 1 subvarieties C_{n} and M_{n} in H_{n}; the first of them is called the caustic and it is the closure of the space of functions having a critical point of order 2, the second one is the Maxwell stratum and it is the closure of the space of functions having two distinct critical points with coinciding critical values.

Conjecture 1.1 The subalgebra of the cohomology algebra H^{∗}(ℙH_{n}) generated by the cohomology classes of the strata is generated by the two classes C_{n} and M_{n} in H^{2}(ℙH_{n}).

Up to now, the basis of this conjecture is not too solid. But it does not contradict our sample calculations, and it has a nonformal justification coming from Kazarian’s theory.

Kazarian’s theory concerns cohomology classes of multisingulatities of a map f : M → N of two complex manifolds, dimM ≤ dimN. Its main statement claims that there is a universal way to express these cohomology classes in terms of the characteristic classes of the tangent bundles over the two manifolds M and N, more precisely in terms of the class c(T M)/f^{∗}c(T N), where c is the total Chern class. It is a development of the theory of the Thom polynomial. Up to now, only a preliminary version [9] of the text describing the theory is available, and we are not going to refer directly to statements from it. However, the ideology of the theory seems to be applicable in the situation we are studying, and it leads to a number of conjectures concerning the part of the cohomology ring of the Hurwitz spaces we are interested in.

In our situation, the variety M is the universal curve U_{n} over the Hurwitz space ℙH_{n}, while the variety N is the quotient of H_{n} × ℂP^{1} by the natural ℂ^{∗} action. The map f is the universal map over ℙH_{n}. It is easy to see that the map f almost identifies the tangent spaces T M and T N; more precisely, one has c(T M)/f^{∗}c(T N) = (1+a)/(1+b), where a and b are first Chern classes of some linear bundles. However, new complications arise, since we are interested in cohomology classes of ℙH_{n} and not of U_{n}.

Kazarian’s ideology leads to Conjecture 1.1.

If the conjecture is true, then the computation of intersection numbers of the strata in the Hurwitz space with the complementary powers of the class Ψ_{n} ∈ H^{2}(ℙH_{n}) can be split into two independent stages:

• express the cohomology class of the required stratum as a (homogeneous) polynomial in the classes M_{n} and C_{n};

• find the intersection index of each monomial in M_{n} and C_{n} with the complementary degree of the class Ψ_{n}.

Another cohomology class in H^{2}(ℙH_{n}) is also distinguished. This is the class Δ_{n} dual to the subvariety of functions defined on singular curves. This class also can be expressed in terms of the classes C_{n} and M_{n}. This means that the subring in H^{∗}(ℙH_{n}) generated by the classes C_{n} and M_{n} coincides with that generated by Ψ_{n} and Δ_{n}. The linear relations relating these four classes are as follows:

C_{n} = 6(n − 1)Ψ_{n} − 3Δ_{n};
M_{n} = 2(n − 1)(n − 6)Ψ_{n} + 4Δ_{n}.

The coefficients in these linear relations are polynomial in n, which (together with the informal support of Kazarian’s theory) allows us to sharpen Conjecture 1.1 in the following way:

Conjecture 1.2 Each cohomology class dual to a stratum in ℙH_{n} can be expressed as a homogeneous polynomial in Ψ_{n}, Δ_{n} whose coefficients are polynomials in n.

Therefore the calculations must even be simplified.

1.4 Plan of the papers

In Section 2 we give precise definitions of notions mentioned in this introduction. Section 3 is devoted to the description of the Lyashko–Looijenga map. In Section 4 the structure of cohomology of the Hurwitz spaces is analyzed.

1.5 Acknowledgments

We are greatful to A. Zorich who invited us both to Rennes, where the main part of this work was accomplished. The work was completed during our participation in the Mini-Workshop on Hurwitz theory and its ramifications, and we thank its organizers J. H. Kwak and A. D. Mednykh for inviting us. We also thank B. Shapiro and M. Shapiro for their interest and M. Kazarian for explaining the basics of his theory.

2 Main definitions

In this section we briefly recall the definitions of stable curves and stable maps (we will restrict our attention to curves of genus 0), as well as the definition of the Hurwitz space H_{n} from [2].

2.1 Moduli spaces of stable curves

Consider the Riemann sphere ℂP^{1} with n ≥ 3 distinct numbered marked points on it. Two choices of marked points are considered equivalent if there exists an automorphism of ℂP^{1} sending one set of marked points to the other, preserving their numbering.

The space of all nonequivalent choices of n numbered marked points on ℂP^{1} can be endowed with a natural structure of a smooth irreducible (noncompact) complex manifold of dimension n − 3. We denote this moduli space by M_{n}.

Example 2.1 The space M_{3} is a point, because any triple of points on ℂP^{1} can be sent to any other triple by a biholomorphic map. Similarly, M_{4} is isomorphic to ℂP^{1}−{0, 1, ∞} (here we suppose that the projective line is endowed with an arbitrary complex coordinate). Indeed, for any four marked points on ℂP^{1} one can send the first three to 0, 1, and ∞. The fourth marked point will be sent to some uniquely determined point λ, different from 0, 1, and ∞, and this value of λ is the point of M_{4}.

The spaces M_{n} are compactified by adding to them new points corresponding to singular stable curves. All the curves under consideration are compact. The only singularities allowed are simple nodes, or double points (as at the origin in the plane curve xy = 0); such curves are called nodal.

Separating the two branches of a nodal curve at each node, we obtain a disjoint union of smooth compact curves, which is called the normalization of the original curve. The connected components of the normalization are the irreducible components of the curve. Associate to a nodal curve a graph whose vertices correspond to the connected components of the normalization, and two vertices are connected by an edge iff the corresponding components intersect at a node (loops appear if a component intersects itself). This graph is called the modular graph of the nodal curve. By definition, the genus of a nodal curve is zero if all connected components of its normalization are projective lines, and the modular graph is a tree.

Definition 2.2 A stable genus 0 curve is a nodal curve of genus 0 with n distinct marked and numbered points satisfying the following conditions: (i) the marked points do not coincide with the nodal points; (ii) the number of automorphisms of the curve preserving the marked points is finite.

The second condition is equivalent to the following easily verifiable condition (ii’): the total number of marked and nodal points on every irreducible component of the nodal curve is greater than or equal to 3.

The space of genus zero stable curves with n marked points is endowed with a natural structure of a smooth irreducible compact complex manifold of dimension n − 3; we denote this space by \overline{M}_{n}. The space M_{n} of smooth curves form a dense subvariety in \overline{M}_{n}.

Example 2.3 Apart from smooth curves, there are exactly 3 stable genus 0 curves with 4 marked points. They are shown in Fig. 1. If λ from Example 2.1 tends to 0

Figure 1: Singular stable curves with 4 marked points.

(respectively, 1 or ∞), the corresponding smooth curve tends to the 3rd (respectively 2nd, or 1st) stable curve in the figure.

Accordingly, \overline{M}_{4} is obtained from M_{4} by adding three initially punctured points. Actually, \overline{M}_{4} is isomorphic to ℂP^{1}.

2.2 Hurwitz spaces

Now consider the space of meromorphic functions on ℂP^{1} with exactly n ≥ 1 numbered simple poles. This space has a structure of a smooth (noncompact) complex manifold of dimension 2n − 2. In the case when n ≥ 3, if we forget the function itself and only remember the positions of its poles, we obtain a map from the space of functions onto M_{n}. Because of this, we will often say “marked points” instead of “poles of f”.

We are going to construct a compactification of the space of meromorphic functions following [2], similar to that of the moduli space M_{n}. To do that, we will need to define meromorphic functions on nodal curves. A meromorphic function f on a nodal curve S is simply a meromorphic function defined on each irreducible component of the nodal curve. If two components intersect at a nodal point, the function must take the same value at this point on both components. Two pairs (S_{1}, f_{1}) and (S_{2}, f_{2}) are equivalent if there exists a biholomorphic isomorphism φ : S_{1} → S_{2} that preserves the numbering of the poles and makes the following diagram commutative:

[figure: S_{1}; S_{2}; ℂP^{1}; φ; f_{1}; f_{2}]

Definition 2.4 A stable genus 0 meromorphic function is a function f : S → ℂP^{1} defined on a nodal curve S and satisfying the following conditions. (i) The function f does not have poles at nodal points. (ii) The number of automorphisms of the pair (S, f) is finite.

As above, a comment must be made about the second condition. An automorphism of the pair (S, f) is a map φ from S to itself that makes the following diagram commutative:

[figure: S; S; ℂP^{1}; φ; f; f]

Condition (ii) is again easy to check, because it is equivalent to (ii’): If the function f is constant on some irreducible component of S, then the number of nodal points on this component is greater than or equal to 3. (Note that such a component cannot contain marked points precisely because f is constant on it.)

Definition 2.5 The Hurwitz space H_{n} (for n ≥ 2) is the space of all stable genus 0 meromorphic functions with n simple numbered poles.

There is a natural action of ℂ^{∗} on H_{n} defined by simply multiplying a stable function by a constant. The invariant points of this action are stable functions such that the nodal curve has at most one pole on every component and the function vanishes on the components without poles.

These points form the zero section of the Hurwitz space. (We will soon see that the Hurwitz space is a vector bundle and the points in question indeed form its zero section.)

Definition 2.6 The projectivized Hurwitz space ℙH_{n} is the quotient of H_{n}, without the zero section, by ℂ^{∗}.

For n ≥ 3, ℙH_{n} has the structure of a smooth compact complex manifold of dimension 2n − 3. ℙH_{2} is a compact smooth orbifold.

Example 2.7 ℙH_{2} is isomorphic (as an orbifold) to the weighted projective line with weights 2 and 1, that is to

(ℂ^{2} − {0})/(z, w) ∼ (λ^{2}z, λw).

Indeed, if we are given a meromorphic map with 2 poles on ℂP^{1}, we can move its poles to 0 and ∞ and write the function as f(z) = az+b+c/z. The numbers a, b, c are considered up to (simultaneous) multiplication by a scalar factor. Moreover, making a change of variables z ↦ λz, we see that the function (λa)z+b+(c/λ)/z is equivalent to f. Thus [ac : b] is a set of homogeneous coordinates in ℙH_{2}. The point ac = 0 corresponds to the stable meromorphic function defined on a nodal curve with two irreducible components.

The space ℙH_{3}, as will soon be proved, is isomorphic to ℂP^{3}.

2.3 H_{n} is a bundle over \overline{M}_{n}

Now we are going to prove that for n ≥ 3, ℙH_{n} is the projectivization of an (n + 1)- dimensional holomorphic vector bundle on \overline{M}_{n}. First let us define several line bundles on \overline{M}_{n}.

Consider a stable curve S with n marked points. For i from 1 to n, let L_{i} and L_{i}^{∨} be the complex line respectively cotangent and tangent to S at the ith marked point.

Definition 2.8 L_{i} (respectively L_{i}^{∨}) is the line bundle over \overline{M}_{n} whose fiber over a point S ∈ \overline{M}_{n} is the line L_{i} (respectively L_{i}^{∨}).

Denote by ℂ the trivial line bundle and let E_{n} momentarily denote the following (n + 1)-dimensional bundle over \overline{M}_{n}:

E_{n} = L_{1}^{∨} + … + L_{n}^{∨} + ℂ.

Proposition 2.9 For n ≥ 3, we have H_{n} = E_{n}.

Proof. We will show that giving a stable meromorphic function on a nodal curve is equivalent to giving a point in the total space of the bundle E_{n}. Considering stable functions up to a scalar factor corresponds to taking the projectivization of E_{n}.

First of all, note that if a meromorphic function f has a simple pole at some smooth point z of a nodal curve S, then it determines a tangent vector to S at z; more precisely, the principal part of the pole is such a tangent vector. Indeed, if we want to couple this tangent vector with a cotangent vector represented by a 1-form ω, the result will be the residue of fω at z.

Now, given a stable meromorphic function on a nodal curve, we can divide its poles into two parts. The poles of the first type lie on the irreducible components of the curve that contain at least 3 marked and double points. The poles of the second type lie on the irreducible components that contain a unique pole and a unique nodal point. (These are the only possible cases.)

If we forget the function itself and retain only the positions of its poles, we obtain a nodal curve with n marked points. This curve is not necessarily stable. However, we can make it stable by contracting every component that contains only 1 nodal and 1 marked point. The former node becomes the marked point. Thus we obtain a map from H_{n} to \overline{M}_{n}.

Further, consider the stable curve that we have obtained. We assign to its marked points of the first type the principal parts of the corresponding poles (they are tangent vectors at the marked points). We assign zero tangent vectors to the marked points of the second type. Finally, we assign to the stable meromorphic function a complex number: the sum of its critical values given by the Lyashko–Looijenga map (see Section 3 below). Thus we have assigned to the stable meromorphic function a point in the total space of E_{n}.

Conversely, given a point in the total space of E_{n}, we can easily recover the stable meromorphic function. First we extend the stable curve by adding new irreducible components at each marked point of the second type (those, whose tangent vector is equal to 0). Then, knowing the principal parts of the poles, we can recover the function f uniquely up to an additive constant. Knowing the sum of its critical values allows us to fix the constant. ⋄

Remark 2.10 The vector bundle H_{n} can be understood more conceptually.

Consider the space S_{n}(ℂP^{1}) of all stable degree n maps from genus 0 curves S to ℂP^{1}. Inside this space there is a suborbifold canonically isomorphic to \overline{M}_{n}/S_{n}, where S_{n} is the symmetric group acting on \overline{M}_{n} by renumbering the poles. This subvariety consists of stable maps of the following form. Take a stable genus 0 curve S′ with n marked but not numbered points. Attach to S′ a new spherical component at each marked point. Instead of the former marked points, mark one point on each of the new spherical components. Thus we have obtained a nodal curve S. Consider the map f : S → ℂP^{1} that sends the whole curve S′ to 0 and is of degree one on each of the new spherical components, the poles being at the new marked points. Such stable maps (S, f) form an orbifold isomorphic to \overline{M}_{n}/S_{n}.

Now, the normal bundle to \overline{M}_{n}/S_{n} in S_{n}(ℂP^{1}) (in the orbifold sense) is canonically identified with the quotient H_{n}/S_{n}.

Definition 2.11 We denote by T the canonical line bundle over ℙH_{n} and by Ψ_{n} = c_{1}(T^{∨}) ∈ H^{2}(ℙH_{n}) the first Chern class of the dual line bundle.

3 The Lyashko–Looijenga map

3.1 The definition of the map

Consider the space of unordered sets of m (not necessarily distinct) complex numbers. This space is isomorphic to the space of monic polynomials of degree m: just take a set of m numbers to the polynomial whose roots are these numbers.

Definition 3.1 The Lyashko–Looijenga map LL is the map that assigns to a meromorphic function with n simple poles on ℂP^{1} the (unordered) set of its 2n − 2 critical values counted with multiplicities.

We recall that a critical point of a function f is a point, where df = 0, while a critical value of f is its value at a critical point. The multiplicity of a critical point is the multiplicity of the zero of df at this point. In other words, if the function f has the form f(z) = z^{k} + c in some local coordinate z, then z = 0 is a critical point of multiplicity k − 1. The multiplicity of a critical value is the sum of multiplicities of the critical points at which the critical value in question is attained. If we consider the function as a ramified covering of ℂP^{1}, then its critical values are exactly those points in the image over which the covering is ramified.

The notion of a cone (or of a cone bundle) generalizes that of a vector bundle to the case of singular fibers, see [3]. The essential point is that cones carry a fiberwise action of the multiplicative group ℂ^{∗} of nonzero complex numbers. A cone morphism preserves this action.

Proposition 3.2 ([2]) The Lyashko–Looijenga map extends to a cone morphism

LL : H_{n} → ℂ^{2n−2}.

The proof of the proposition is based on the extension of the notion of set of critical values to maps defined on nodal curves.

First consider all the irreducible components of S where f is constant. Consider the union of these irreducible components, and let S_{0} be a connected component of this union. (In general S_{0} consists of several irreducible components intersecting at double points.) Suppose S_{0} contains k nodal points at which it intersects other irreducible components (on which f is not constant). Then the value of f on S_{0} is considered a critical value of multiplicity 2k − 2.

Second, consider a nodal point lying on two irreducible components of S such that f is not constant on either of them. The value that f takes at such a point is considered a double critical value.

Finally, the critical values of f on all the irreducible components are taken into account as ordinary critical values.

One can prove that in that way we have obtained a set of 2n − 2 critical values, and that this set depends continuously on the point of H_{n}. See [2] for more details.

3.2 A stratification of the Hurwitz space

Consider a degree n ramified covering f : S → S^{2} of the sphere. Let t ∈ S^{2} be a ramification point of f of multiplicity d. To this ramification point we can assign a partition κ of d. This partition consists of the multiplicities of all critical points of f in S whose image under f is equal to t. We will write such a partition in the multiplicative form, κ = 1^{m_{1}} … d^{m_{d}}, where m_{i}, i = 1, …, d is the number of critical points in the preimage of t having multiplicity i, 1 · m_{1} + … + d · m_{d} = d(κ) = d. If the covering has c ramification points, t_{1}, …, t_{c}, then c partitions κ_{1}, …, κ_{c} are associated to it.

Sets of partitions of n that can appear as sets of ramification partitions of a ramified covering are subject to a number of constraints. If the covering surface S also is the sphere, the Riemann–Hurwitz formula implies that the total degeneracy of the set of ramification partitions of a meromorphic function must be 2n − 2,

d(κ_{1}) + … + d(κ_{c}) = 2n − 2.

However, this property alone does not guarantee the existence of a ramified covering with the prescribed ramification type.

Definition 3.3 The set of all rational functions on ℂP^{1} with n simple numbered poles such that the set of partitions assigned to their critical values is {κ_{1}, …, κ_{c}} is called an open stratum. Its closure in H_{n} is called just a stratum and denoted by Σ_{{κ_{1},...,κ_{c}}}.

The projectivizations of these strata (lying in ℙH_{n}) are defined in the obvious way. They will be denoted by ℙΣ_{{κ_{1},...,κ_{c}}}.

Sometimes we will omit reference to simple partitions κ = 1 in the index, omitting in this case the braces as well.

In the sequel we will often describe strata by specifying the set of functions that belong to their open parts and omitting the words “closure of”. The reader should bear in mind that we always consider closed strata, unless explicitly stated otherwise.

Example 3.4 There are two strata of complex codimension 1. They are the caustic C_{n} ⊂ H_{n} and the Maxwell stratum M_{n} ⊂ H_{n}. The caustic C_{n} = Σ_{2} is (the closure of) the set of functions with a double critical point. The Maxwell stratum M_{n} = Σ_{1^{2}} is (the closure of) the set of functions with the same critical value taken at two distinct critical points.

Definition 3.5 The degree of the LL map on a stratum Σ_{{κ_{1},...,κ_{c}}} is the number of the preimages in Σ_{{κ_{1},...,κ_{c}}} of a generic point in LL(Σ_{{κ_{1},...,κ_{c}}}). The degree is denoted by μ_{{κ_{1},...,κ_{c}}}.

Example 3.6 Since the space \overline{M}_{3} is a point, the space H_{3} is isomorphic to ℂ^{4} = L_{1}^{∨} + L_{2}^{∨} + L_{3}^{∨} + ℂ, where the L_{i} are the cotangent lines at the marked points on ℂP^{1}. It can be identified with the space of rational functions of the form

az − \frac{b}{z} − \frac{c}{z − 1} + d.

In Figure 2 we represent the stratification of the projectivized Hurwitz space ℙH_{3}. The figure shows the real part of this space with a slight simplification: Instead of a 3-dimensional projective space we have represented a 2-dimensional projective space, dropping the parameter d.

The interesting subvarieties in ℙH_{3} are the following ones. First, the projectivized caustic ℙC_{3}, which is a singular cubic with one self-intersection point. It has the homogeneous equation

a^{3} + b^{3} + c^{3} + 3ab^{2} + 3ac^{2} + 3ba^{2} + 3bc^{2} + 3ca^{2} + 3cb^{2} − 21abc = 0.

Since our picture shows the real part of ℙH_{3}, the self-intersection point, which has the homogeneous coordinates (1 : 1 : 1), appears to lie apart from the rest of the cubic. Second, the three projective lines a = 0, b = 0, c = 0, which correspond to reducible curves with one pole lying on one component and the other two poles on the other one. The Maxwell stratum is empty. The cubic intersects each of the three lines at one point, but the intersection has multiplicity three. These three points lie on the line a + b + c = 0.

The boxes around the picture of ℙH_{3} show, for each of the subvarieties, what are the corresponding stable rational functions. Short traits represent poles, crosses represent critical points, while crosses with double lines represent double critical points. The numbers written near the arrows are the multiplicities of the LL map on the corresponding subvariety.

Now we are able to prove Theorem 4 from the introduction. We restate it here. Consider a stratum Σ_{{κ_{1},...,κ_{c}}} ⊂ H_{n} (see Definitions 3.3 and 3.5) determined by

partitions κ_{1}, …, κ_{c}. Let d_{1} = d(κ_{1}), …, d_{c} = d(κ_{c}) be the multiplicities of the critical values. Let |Aut{d_{1}, …, d_{c}}| be the number of automorphisms of the set {d_{1}, …, d_{c}}, i.e., the number of permutations σ of c elements such that d_{i} = d_{σ(i)} for all i.

Theorem 4 The degree μ_{{κ_{1},...,κ_{c}}} of the Lyashko–Looijenga map on Σ_{{κ_{1},...,κ_{c}}} equals

|Aut{d_{1}, …, d_{c}}| ⟨ℙΣ_{{κ_{1},...,κ_{c}}}, Ψ_{n}^{d}⟩,

where d is the dimension of the stratum ℙΣ_{{κ_{1},...,κ_{c}}} and ⟨·, ·⟩ is the coupling between a homology and a cohomology class.

Figure 2: The stratification of the “simplified” projectivized space ℙH_{3}.

Proof of Theorem 4. To prove the theorem we consider a (2n − 2)!-sheeted ramified covering ℙ̃H_{n} of ℙH_{n}. By definition, a point of ℙ̃H_{n} corresponds to a stable meromorphic function with numbered critical points.

More precisely, consider the Zariski open subset X of ℙH_{n} constituted by functions f defined on ℂP^{1}, with 2n − 2 simple critical points. Over this open subset we can construct a (2n − 2)!-sheeted non-ramified covering X̃, whose points correspond to functions with numbered critical points. It is a connected complex manifold. Then this covering is uniquely extended to a ramified covering of ℙH_{n} in the following way.

For simplicity we start by describing the set of preimages of every point f_{0} ∈ ℙH_{n} in the covering ℙ̃H_{n}. Consider an f_{0} ∈ ℙH_{n} and a small ball B surrounding f_{0}. A point f ∈ B ∩ X has (2n − 2)! preimages in the covering. The local monodromy group π_{1}(X ∩B) acts by permutations on these preimages. The preimages of f_{0} in the covering are the orbits of this action. Now let us give an algebro-geometric definition.

Consider an affine chart U in ℙH_{n} and let A be the algebra of algebraic functions on U, U = spec A. Let F ∈ A, be a function whose set of zeroes is the ramification divisor. Then the localization A_{F} is the algebra of algebraic functions on the complement U ∩ X to the ramification divisor. The map A ↪ A_{F} is an injection, because A has no zero divisors. Let B be the algebra of algebraic functions on the lifting { \atop U ∩ X} of U ∩ X on X̃. Then A_{F} ↪ B is again an injection. The composition of the two injections gives an injection A ↪ B. We denote by C the integral closure of A in B. By definition, the ramified covering over U is then the variety Spec C. Such ramified coverings of affine charts can then be glued into a covering of the total space ℙH_{n}.

Lemma 3.7 The variety ℙ̃H_{n} is normal.

Proof. Recall that a normal algebraic variety is a variety such that each of its points possesses an affine neighborhood whose algebra of functions is integrally closed in its field of fractions (see [4]). On normal varieties Chern classes of vector bundles and local intersection indices of subvarieties are well-defined. Normality is a local property, therefore we can restrict our attention to an affine chart U as above.

Let us prove that C is integrally closed. First of all, B is integrally closed, because it is an algebra of functions on a smooth variety. Thus an element x of the field of fractions of C integral over C belongs to B. But C is integrally closed in B (being the integral closure of A), thus x necessarily belongs to C. ⋄

In general, ℙ̃H_{n} is not smooth. For example, the 2-sheeted ramified covering of ℂ^{2} ramified over the axes x = 0 and y = 0 is the cone given by the equation z^{2} = xy.

Our description of the preimage of a point in ℙH_{n} under the projection ℙ̃H_{n} → ℙH_{n} follows from the fact that a normal complex algebraic variety is locally irreducible as a complex manifold ([4], Theorem 6.6).

We denote by L̃L the map

L̃L : ℙ̃H_{n} → ℂP^{2n−3}

that assigns to a meromorphic function with numbered critical points in ℙ̃H_{n} the set of its numbered critical values. (Recall that both the function and the set of its critical values are considered up to a common scalar factor.) Denote by O(−1) the tautological line bundle over the target space ℂP^{2n−3}.

Recall that T is the tautological line bundle over ℙH_{n}. Denote by T̃ the pull-back of T under the projection ℙ̃H_{n} → ℙH_{n}. Then T̃ coincides with the pull-back of O(−1) under L̃L. Indeed, consider a point of the total space of the bundle T̃, outside its zero section. It corresponds to a nonvanishing meromorphic function with numbered critical points (and, this time, not considered up to a scalar factor). Similarly, the total space of the bundle O(−1) without the zero section is ℂ^{2n−2} − {0}, and it corresponds to sets of 2n − 2 numbered critical values, not all of which are equal to 0. It follows that the map L̃L can be lifted to the total spaces of the bundles T̃ and O(−1): it suffices to assign to a meromorphic function the set of its critical values. This lifting identifies the fibers of the two bundles.

It follows that for any subvariety ℙ̃Σ ⊂ ℙ̃H_{n} of pure dimension d such that its image L̃L(Σ̃) is irreducible and also of dimension d, we have

⟨ℙ̃Σ, (c_{1}(T̃^{∨}))^{d}⟩ = μ(L̃L|ℙ̃Σ) deg L̃L(ℙ̃Σ).

Indeed, we have

deg L̃L(ℙ̃Σ) = L̃L(ℙ̃Σ) ∩ P,

where P ⊂ ℂP^{2n−3} is a generic projective subspace of dimension complementary to that of L̃L(ℙ̃Σ). Now, P is the intersection of d projective hyperplanes. A hyperplane is a section of O(1) (the line bundle dual to the tautological line bundle O(−1)).

Let us count in two different ways the number of preimages in ℙ̃Σ of the points of intersection between L̃L(ℙ̃Σ) and P. First, since each intersection point has μ(L̃L|ℙ̃Σ) preimages, the number of preimages equals

μ(L̃L|ℙ̃Σ) deg L̃L(ℙ̃Σ).

Second, the pull-back to ℙ̃H_{n} of a generic hyperplane is a generic section of the line bundle T̃^{∨} dual to T̃. Taking the intersection between d sections like that and the subvariety ℙ̃Σ, we obtain

⟨ℙ̃Σ, (c_{1}(T̃^{∨}))^{d}⟩ .

Hence, these two numbers are equal.

The same equality holds if L̃L(ℙ̃Σ) is not irreducible, but all generic points of L̃L(ℙ̃Σ) have the same number of preimages.

Now let ℙ̃Σ be the lifting to ℙ̃H_{n} of a stratum ℙΣ_{{κ_{1},...,κ_{c}}} in ℙH_{n}. Denote by d their common dimension. For a partition κ denote by κ! the product of the factorials of its elements. It is easy to see that a generic point in ℙΣ_{{κ_{1},...,κ_{c}}} has

\frac{(2n − 2)!}{κ_{1}! … κ_{c}!}

liftings to Σ̃. Further, the image of ℙ̃Σ under the lifted Lyashko–Looijenga map L̃L is a union of

\frac{(2n − 2)!}{d_{1}! … d_{c}! |Aut\{d_{1}, …, d_{c}\}|}

projective subspaces of dimension d.

Finally, denote by

μ̃_{{κ_{1},...,κ_{c}}} = μ(L̃L|_{ℙ̃Σ_{{κ_{1},...,κc}}})

the degree of L̃L on ℙ̃Σ_{{κ_{1},...,κ_{c}}}, which is the number of preimages in ℙ̃Σ_{{κ_{1},...,κ_{c}}} of a generic point in the image of ℙ̃Σ_{{κ_{1},...,κ_{c}}}. Simple combinatorial considerations show that

μ̃_{\{κ_{1},...,κ_{c}\}} = ∏_{i=1}^{c} \frac{d_{i}!}{κ_{i}!} · μ_{\{κ_{1},...,κ_{c}\}}.

The factor ∏ \frac{d_{i}!}{κ_{i}!} is the number of ways to number the critical points once the critical values are numbered.

Putting everything together we see that

μ_{\{κ_{1},...,κ_{c}\}} = ∏_{i=1}^{c} \frac{κ_{i}!}{d_{i}!} · μ̃_{\{κ_{1},...,κ_{c}\}}

= \frac{∏ κ_{i}!}{∏ d_{i}!} · \frac{∏ d_{i}! |Aut\{d_{1}, …, d_{c}\}|}{(2n − 2)!} · ⟨ℙ̃Σ, (c_{1}(T̃^{∨}))^{d}⟩

= |Aut{d_{1}, …, d_{c}}| ⟨ℙΣ_{{κ_{1},...,κ_{c}}} , (c_{1}(T^{∨}))^{d}⟩

= |Aut{d_{1}, …, d_{c}}| ⟨ℙΣ_{{κ_{1},...,κ_{c}}} , Ψ_{n}^{d}⟩ .

This proves the theorem. ⋄

4 The cohomology of the Hurwitz space

In this section we describe the cohomology algebra of the moduli spaces and of the Hurwitz spaces. We also prove several cohomological identities which will be used in the subsequent paper [15] to derive recurrence relations on Hurwitz numbers.

4.1 Keel’s description of H^{∗}(\overline{M}_{n})

The description of the cohomology algebra of the space \overline{M}_{n} given in this section is due to Keel [10].

Proposition 4.1 ([10]) The cohomology ring H^{∗}(\overline{M}_{n}) is generated by H^{2}(\overline{M}_{n}).

Now we will describe a set of 2-cohomology classes that span H^{2}(\overline{M}_{n}). (More precisely, we will describe cycles that represent their Poincaré dual homology classes.)

Denote by D = A ⊔ B an unordered partition of the set {1, …, n} of marked points into two disjoint parts (a partition A ⊔ B coincides with B ⊔ A). Each part must contain at least 2 points.

Consider all stable curves with exactly two irreducible components such that one component contains the marked points of the set A and the other one of the set B. Such stable curves form a (complex) codimension 1 subvariety of \overline{M}_{n}. Its closure is a smooth closed subvariety of \overline{M}_{n} of complex codimension 1. Therefore it represents a 2-cohomology class which we will denote by [D].

Proposition 4.2 ([10]) The classes [D] span H^{2}(\overline{M}_{n}).

The generators [D] are not linearly independent. For example, for n = 4 all the three generators corresponding to the partitions

D_{1} = {1, 2} ⊔ {3, 4} , D_{2} = {1, 3} ⊔ {2, 4} , D_{3} = {1, 4} ⊔ {2, 3}

represent the same cohomology class, (Poincaré dual to) a point.

More generally, let us fix four distinct numbers i, j, k, l between 1 and n. Denote by [ijDkl] the sum of all the generators [D] ∈ H^{2}(\overline{M}_{n}) corresponding to partitions, where i and j belong to one of the two parts, while k and l belong to the other part. Then the generators [D] satisfy the relations

R_{ijkl} : [ijDkl] = [ikDjl] = [ilDjk].

These relations can be deduced using the forgetful map \overline{M}_{n} → \overline{M}_{4} which forgets all the marked points except for i, j, k, l and contracts all the components of the curve that have become unstable.

Proposition 4.3 ([10]) The relations R_{ijkl} span all the linear relations on the generators [D].

Finally, we say that two partitions D_{1} and D_{2} are compatible if the set {1, …, n} can be divided into a disjoint union of three sets A, B, C, in such a way that

D_{1} = (A ⊔ B) ⊔ C, D_{2} = A ⊔ (B ⊔ C).

It is easy to see that if two partitions D_{1} and D_{2} are not compatible, then the geometric intersection of the corresponding cycles has codimension greater than 2. Therefore, the intersection of two such generators in the cohomology algebra vanishes.

Theorem 5 ([10]) The cohomology algebra H^{∗}(\overline{M}_{n}) is generated by the generators [D] modulo two kinds of relations: (i) the linear relations R_{ijkl}, (ii) the multiplicative relations [D_{1}][D_{2}] = 0 for incompatible partitions D_{1} and D_{2}.

This theorem gives a complete description of the cohomology algebra of \overline{M}_{n}, although in practice it leads to rather heavy computations.

Using this theorem, Kontsevich and Manin [11] found linear generators and relations of H^{k}(\overline{M}_{n}) for all k.

4.2 A description of the cohomology of Hurwitz spaces

In what follows, we will require particular 2-cohomology classes ψ_{i} ∈ H^{2}(\overline{M}_{n}). Recall that L_{i} denotes the line bundle over \overline{M}_{n} whose fiber coincides with the cotangent line at the ith marked point.

Definition 4.4 We denote by ψ_{i} = c_{1}(L_{i}) the first Chern class of the line bundle L_{i}.

For n ≥ 6 the classes ψ_{i} do not span H^{2}(\overline{M}_{n}). The following proposition expresses them in terms of the classes [D].

Let i, j, k ∈ {1, …, n}. We denote by [i ∗ Djk] the sum of all the classes [D] corresponding to partitions containing i in one part, and j, k in the other part. The star in the notation reminds that an irreducible component with a single node cannot contain a single marked point.

Proposition 4.5 For each i and any distinct j, k different from i we have ψ_{i} = [i ∗ Djk].

Proof. We construct a holomorphic section of the line bundle L_{i} in the following way. Consider a stable curve such that the marked points i, j, k are on the same irreducible component. On this component there exists a unique meromorphic differential with simple poles at the points j and k, whose residues at these poles are equal to 1 and −1 respectively. The value of this differential at the point i is an element of the line L_{i}. Thus we obtain a section of the bundle L_{i}, and it extends in a unique way to a meromorphic section over the entire base \overline{M}_{n}. It is easy to see that this section has no poles (thus the section is, actually, holomorphic). Its zeroes are precisely the classes that add up to i ∗ Djk, and it intersects the zero section of L_{i} transversally. ⋄

Since H_{n} is a vector bundle over \overline{M}_{n}, the cohomology algebra of \overline{M}_{n} can be canonically seen as a subalgebra of the cohomology algebra of ℙH_{n}. We will therefore use the same notation for cohomology classes on \overline{M}_{n} and their pull-backs on ℙH_{n}.

Now we express the cohomology algebra of ℙH_{n} in terms of that of \overline{M}_{n}.

Theorem 6

H^{∗}(ℙH_{n}) = H^{∗}(\overline{M}_{n})[Ψ_{n}] / (Ψ_{n} − ψ_{1}) … (Ψ_{n} − ψ_{n})Ψ_{n}.

Proof. The assertion follows immediately from the description of H_{n} as a vector bundle over \overline{M}_{n} (Section 2.3) and from the following well-known fact (see, for example, [3]). For any rank n complex vector bundle E over a complex manifold M we have

H^{∗}(ℙE) = H^{∗}(M)[Ψ] / (Ψ^{n} + c_{1}(E)Ψ^{n−1} + … + c_{n}(E)),

that is, the cohomology algebra of ℙE is isomorphic to the algebra of polynomials in one variable Ψ with coefficients in H^{∗}(M), modulo the Chern polynomial in Ψ. Here the new variable Ψ can be identified with the first Chern class of the tautological line bundle over ℙE. ⋄

This gives an explicit description of the cohomology algebra of ℙH_{n} for n ≥ 3.

Now, let D = A ⊔ B be a partition of the set {1, …, n} into a disjoint union of two nonempty subsets. The subsets are no longer required to have at least 2 elements. Consider the set of stable meromorphic functions defined on nodal curves with 2 irreducible components, the first component containing the marked points of the subset A and the second one of the subset B. This set is an open submanifold of H_{n} of complex codimension 1, so its closure is a cycle Poincaré dual to a cohomology class in H^{2}(ℙH_{n}). This class will be denoted by [D].

If A and B both contain at least two elements, then the corresponding class [D] is a pull-back from a 2-cohomology class on \overline{M}_{n}.

Let i, j, k ∈ {1, …, n}. Denote by [iDjk] ∈ H^{2}(ℙH_{n}) the sum of all the classes [D] corresponding to partitions such that i belongs to one part and j and k belong to the other one. (There is no star after i, because the singleton {i} is now an allowed subset.)

Proposition 4.6 For any pairwise distinct i, j, k ∈ {1, …, n}, we have Ψ_{n} = [iDjk].

Proof. Recall that Ψ_{n} is the first Chern class of the tautological line bundle on ℙH_{n}. A section of the vector bundle H_{n}^{∨} dual to H_{n} will automatically provide a section of O_{s}(−1) (because a linear form on each fiber of H_{n} can be restricted to any line contained in that fiber). But a section of the bundle H_{n}^{∨} can be obtained from a section of any of the line bundles L_{i} by taking its direct sum with zero sections of each of the other line bundles L_{j} for j ≠ i. Thus the class Ψ_{n} is equal to the sum of the class ψ_{i} = c_{1}(L_{i}) and of the class Poincaré dual to the projectivization of the subbundle

L_{1}^{∨} + … + L_{i−1}^{∨} + L_{i+1}^{∨} + … + L_{n}^{∨} + ℂ.

The latter 2-cohomology class is exactly the class that corresponds to the partition

D = {i} ⊔ {1, …, i − 1, i + 1, … n}.

Using the expression for ψ_{i} given in Proposition 4.5 we obtain Ψ_{n} = [iDjk]. ⋄

Proposition 4.7 For n ≥ 2, the classes [D] span H^{2}(ℙH_{n}) and generate H^{∗}(ℙH_{n}) as an algebra.

Proof. The first assertion clearly follows from the second one. Let us prove the second assertion. For n = 2, ℙH_{2} is topologically a sphere, and the class [D] = [{1} ⊔ {2}] is a point; thus the proposition is true. For n ≥ 3, the classes [D] corresponding to partitions with at least two elements in each part generate the cohomology algebra of \overline{M}_{n}, and those corresponding to partitions with a single element in one part produce the class Ψ_{n}. Therefore, they generate the cohomology algebra of ℙH_{n}. ⋄

4.3 Cohomological identities on the Hurwitz spaces

In this section we study various relations between the cohomology classes Poincaré dual to strata in ℙH_{n}. Note that the symmetric group S_{n} acts on H_{n} by permuting the poles. Therefore, it also acts on the cohomology algebra of ℙH_{n}. Strata of the stratification, as well as most of other cohomology classes that we consider, are invariant under this action. Therefore, their cohomology classes belong to the subalgebra of S_{n}-invariant cohomology classes.

Recall that C_{n} (respectively M_{n}) denotes the 2-cohomology class Poincaré dual to the caustic (respectively the Maxwell stratum) in ℙH_{n} (see Example 3.4).

Let p and q be two positive integers, p, q ≥ 1, p + q = n. We have introduced the cohomology class [A ⊔ B] assigned to a partition A ⊔ B of the set {1, …, n} into two nonempty parts. Denote by Δ_{p,q} the sum of the classes [A ⊔ B] for all partitions such that |A| = p, |B| = q. (If p = q the partitions A ⊔ B and B ⊔ A are two different terms of the sum, although they determine the same cohomology class.)

Definition 4.8 Denote by Δ_{n} the 2-cohomology class

Δ_{n} = \frac{1}{2} ∑_{p+q=n} Δ_{p,q}

and call it the boundary stratum.

The classes introduced above satisfy the following linear relations.

Proposition 4.9 We have

Ψ_{n} = \frac{1}{2n(n − 1)} ∑_{p+q=n} pq Δ_{p,q};

Ψ_{n} = \frac{1}{(2n − 2)(2n − 3)}(3C_{n} + 2M_{n} + Δ_{n}).

Proposition 4.10

C_{n} = 6(n − 1)Ψ_{n} − 3Δ_{n}.
M_{n} = 2(n − 1)(n − 6)Ψ_{n} + 4Δ_{n}.
C_{n} = 3 ∑_{p+q=n} (\frac{1}{n} pq − \frac{1}{2}) Δ_{p,q} ,

M_{n} = ∑_{p+q=n} (\frac{n − 6}{n} pq + 2) Δ_{p,q} .

Proof of Proposition 4.9. The first identity follows from the identity of Proposition 4.6,

Ψ_{n} = [iDjk],

by summing it over all triples (i, j, k) and regrouping the terms.

In order to prove the second identity we consider once again the covering ℙ̃H_{n} of ℙH_{n} introduced in the proof of Theorem 4. Denote by Ψ̃_{n} ∈ H^{2}(ℙ̃H_{n}) the pull-back of the class Ψ_{n} on ℙH_{n}. In the proof of Theorem 4 we established that

c_{1}(L̃L^{∗}(O(−1))) = Ψ̃_{n}.

(The pull-back under the lifted Lyashko–Looijenga map of the class of a hyperplane in ℂP^{2n−3} equals Ψ̃_{n}.) Consider a particular union of hyperplanes in the image space ℂP^{2n−3} of L̃L. Namely, the union of all hyperplanes where the ith and the jth critical values of f (i.e., the ith and the jth coordinates in ℂP^{2n−3}) are equal (for 1 ≤ i < j ≤ 2n − 2). This union represents (2n − 2)(2n − 3)/2 times the class of a hyperplane. Now consider the preimage of this union under the map L̃L. It consists of the union of the strata C̃_{n}, M̃_{n}, and Δ̃_{n} (the liftings to ℙ̃H_{n} of the caustic C_{n}, the Maxwell stratum M_{n}, and the boundary stratum Δ_{n}). Moreover, M̃_{n}, Δ̃_{n} are simple preimages, but C̃_{n} is a triple preimage. (This means that if we take a generic point f ∈ C̃_{n} and its image in ℂP^{2n−3}, and then take a generic point close to the image, then this point will have three preimages close to f.) Now, the projection of ℙ̃H_{n} onto ℙH_{n} is ramified over C_{n} and Δ_{n}, and the order of ramification is 2 in both cases. On the contrary, it is not ramified over M_{n}. This implies that the geometrical liftings of the strata C_{n} and Δ_{n} to ℙ̃H_{n} represent only one half of the pull-backs of the corresponding homology classes. On the other hand, the homology class of the geometrical lifting of M_{n} is equal to the lifting of the homology class of M_{n}. Putting everything together we obtain

\frac{(2n − 2)(2n − 3)}{2} Ψ̃_{n} = 3C̃_{n} + M̃_{n} + Δ̃_{n},

whence

Ψ_{n} = \frac{2}{(2n − 2)(2n − 3)} (\frac{1}{2}(3C_{n})+M_{n}+ \frac{1}{2}Δ_{n}) = \frac{1}{(2n − 2)(2n − 3)} (3C_{n}+2M_{n}+Δ_{n}).

⋄

Proof of Proposition 4.10. The proof of this proposition is surprisingly difficult and uses some results from the subsequent paper [15].

First of all, note that it suffices to prove the first of the four identities C_{n} = 6(n − 1)Ψ_{n} − 3Δ_{n}. The other three identities follow from this one and from Proposition 4.9. Moreover, the 2-cohomology classes in the left- and the right-hand sides of the identity are symmetric, i.e., invariant under the action of the symmetric group S_{n} by renumbering the poles. The symmetric part of H^{2}(ℙH_{n}) is spanned by the classes Δ_{p,q} (see Section 4.2). Therefore, we must construct a set of 2-homology classes of ℙH_{n} such that their couplings with the classes Δ_{p,q} allow one to distinguish all linear combinations of the classes Δ_{p,q}. It is then enough to show that their couplings with C_{n} and 6(n − 1)Ψ_{n} − 3Δ_{n} coincide. To construct such a set of 2-homology classes, we consider several embeddings of ℂP^{1} into ℙH_{n}.

For any positive integers p, q such that p + q = n we are going to consider the following embedding of ℂP^{1} in ℙH_{n}. Let [u : v] be the homogeneous coordinates on ℂP^{1}. Consider the following family of rational functions in the variable z:

f_{u,v}(z) = \frac{1}{uz − a_{1}} + … + \frac{1}{uz − a_{p}} + \frac{1}{vz − b_{1}} + … + \frac{1}{vz − b_{q}} ,

where the a_{i} and the b_{j} are generically chosen complex numbers. The rational functions of this family do not belong to H_{n} for u = 0, v = 0 or a_{i}/u = b_{j}/v (since they have less than n poles); however, there is a unique way to extend the family to a well-defined map from ℂP^{1} to ℙH_{n}, because ℙH_{n} is compact. We denote by σ_{p,q} the 2-homology class of the image of this map. We are going to study its couplings with the 2-cohomology classes C_{n}, Δ_{n}, and Δ_{p,q}. Note that these classes are defined as Poincaré dual classes to particular codimension 1 subvarieties of ℙH_{n}. Therefore we will usually speak of their intersection indices (rather than couplings) with the families Δ_{p,q}.

The three following lemmas describe completely the intersection indices of the family σ_{p,q} with the classes Δ_{p′,q′} and C_{n}. For shortness, we use the following convention. If in a lemma we give the intersection indices σ_{p,q} ∩ Δ_{p′,q′} and σ_{p,q} ∩ Δ_{p′′,q′′}, then in the particular case p′ = p′′, q′ = q′′, the two intersection indices must be added. For example, when we write σ_{p,q} ∩ Δ_{p,q} = σ_{p,q} ∩ Δ_{q,p} = 1, in the particular case p = q the corresponding intersection index is equal to 2.

Lemma 4.11 At each of the points u = 0 and v = 0 we have the following intersection indices:

σ_{p,q} ∩ Δ_{p,q} = σ_{p,q} ∩ Δ_{q,p} = 1;
σ_{p,q} ∩ Δ_{p′,q′} = 0

for other pairs (p′, q′);

σ_{p,q} ∩ C_{n} = 0.

Lemma 4.12 At each of the points u/v = a_{i}/b_{j} we have the following intersection indices:

σ_{p,q} ∩ Δ_{1,n−1} = σ_{p,q} ∩ Δ_{n−1,1} = n − 2;

Figure 3: When u = 0 or v = 0 the function is defined on a 2-component curve with p poles on one component and q on the other one.

σ_{p,q} ∩ Δ_{2,n−2} = σ_{p,q} ∩ Δ_{n−2,2} = 1;
σ_{p,q} ∩ Δ_{p′,q′} = 0

for other pairs (p′, q′);

σ_{p,q} ∩ C_{n} = 3(n − 2).

Lemma 4.13 There are 6(pq − 1) more points of simple intersection of the family σ_{p,q} with C_{n} corresponding to rational functions with a double critical point, defined on a one-component curve.

Before proving the lemmas, let us make use of them to compute the total intersection indices. (For Ψ_{n} we use the expression from Proposition 4.9.) We have

σ_{p,q} ∩ Δ_{1,n−1} = σ_{p,q} ∩ Δ_{n−1,1} = pq(n − 2);
σ_{p,q} ∩ Δ_{2,n−2} = σ_{p,q} ∩ Δ_{n−2,2} = pq;
σ_{p,q} ∩ Δ_{p,q} = σ_{p,q} ∩ Δ_{q,p} = 2;
σ_{p,q} ∩ Δ_{p′,q′} = 0

for other pairs (p′, q′). And

σ_{p,q} ∩ Δ_{n} = pq(n − 1) + 2,  σ_{p,q} ∩ Ψ_{n} = pq,  σ_{p,q} ∩ C_{n} = 3pq(n − 1) − 6.

This is consistent with the identity C_{n} = 6(n − 1)Ψ_{n} − 3Δ_{n} that we must prove.

Moreover, looking at the intersection indices with the classes Δ_{p,q} we conclude immediately that the families σ_{p,q} for different p and q allow one to distinguish any linear combinations of the classes Δ_{p,q}. Therefore, the identity follows from the three lemmas, that we will now prove.

Proof of Lemma 4.11. Consider the case u = 0 (the case v = 0 is similar). When u = 0 the function f is defined on a 2-component curve shown in Figure 3.

Such a stable function belongs to the stratum Δ_{p,q} = Δ_{q,p}, but not to other strata Δ_{p′,q′}, nor to the caustic C_{n}. It is easy to check that the intersection with Δ_{p,q} = Δ_{q,p} is simple (unless p = q, in which case it is double by the definition of Δ_{p,p}). ⋄ Figure 4: When u/v = a_{i}/b_{j} we must multiply the family by ub_{j} − va_{i} in order to get a finite limit in H_{n} (the limit in ℙH_{n} does not change). Thus the principle part of n − 2 poles tend simultaneously to zero, which explains that there appear n − 2 small spheres attached to the central component.

Proof of Lemma 4.12. When u/v = a_{i}/b_{j} the function f is defined on a curve with n irreducible components shown in Figure 4. Such a stable function belongs to n − 2 different irreducible components of Δ_{1,n−1} = Δ_{n−1,1} and to one irreducible component of Δ_{2,n−2} = Δ_{n−2,2}. It is easy to check that the intersection with each component is simple.

Now we must find the index of intersection of the family σ_{p,q} with the caustic C_{n} at the point that we are considering. This is more difficult, because this point does not belong to the smooth part of the caustic. In order to find the intersection index we have to use a result of the subsequent paper [15], Theorem 3.9. In the particular case under consideration the assertion of this theorem is as follows. Consider the subvariety of ℙH_{n} obtained from the stable function f by moving in all possible ways the n − 1 points of intersection of the peripheral components with the central component of the curve in Figure 4, without changing the restrictions of f to the peripheral components. The closure of this subvariety is isomorphic to \overline{M}_{n−1}. Now, the theorem says that the neighborhood of this subvariety in ℙH_{n} is isomorphic to the neighborhood of \overline{M}_{n−1} × \{0\} in H_{n−1} × ℂ^{2}. Moreover, the intersection of the caustic C_{n} with this neighborhood is isomorphic to C_{n−1}×ℂ^{2}, where C_{n−1} is the (nonprojectivized) caustic in the smaller Hurwitz space. The intersection of the family σ_{p,q} with the neighborhood is a generic complex curve that intersects the subvariety \overline{M}_{n−1} transversally and is not contained in C_{n−1} × ℂ^{2}. Now we claim that the intersection of the caustic C_{n−1} with a generic fiber of H_{n−1} over \overline{M}_{n−1} is a hypersurface of degree 3(n − 3), which is therefore the index of intersection between σ_{p,q} and C_{n} at the point under consideration.

It remains to check that 3(n − 3) is indeed the degree of the intersection of C_{n−1} with a generic fiber of H_{n−1} over \overline{M}_{n−1}. To do that it suffices to find the degree of

\frac{1}{c_{1} … c_{n−1}} discrim_{z} (numer (\frac{d}{dz} (\frac{c_{1}}{z − z_{1}} + … + \frac{c_{n−1}}{z − z_{n−1}} )))

as a polynomial in variables c_{1}, …, c_{n−1} (discrim means discriminant, numer means numerator). Indeed, this polynomial is precisely the equation of the caustic C_{n−1} in the fiber, where the poles are fixed at the points z_{1}, …, z_{n−1}. The division by c_{1} … c_{n−1} is needed because, as one can easily check, the hyperplanes c_{i} = 0 are simple zeroes of the discriminant, but do not belong to the caustic. The degree of the above polynomial is indeed equal to 3(n − 3) for homogeneity reasons. ⋄

Proof of Lemma 4.13. In order to calculate the intersection of C_{n} with the family σ_{p,q} outside the points where some poles get glued together, we will consider the discriminant of the derivative of the function f of our family

discrim_{z} (numer (\frac{d}{dz} (\frac{1}{uz − a_{1}} + … + \frac{1}{uz − a_{p}} + \frac{1}{vz − b_{1}} + … + \frac{1}{vz − b_{q}} ))) .

The degree of this discriminant as a homogeneous polynomial in u and v equals 2n(2n − 3). One can check that this polynomial has double zeroes at the points u/v = a_{i}/b_{j}. Moreover, it has a zero of multiplicity 4p^{2} − 6p + 3 at u = 0 and a zero of multiplicity 4q^{2} − 6q + 3 at v = 0. Subtracting the multiplicities of all these zeroes, we obtain 6(pq − 1) zeroes outside the points u = 0, v = 0, u/v = a_{i}/b_{j}. ⋄

Thus the three lemmas are proved, which completes the proof of Proposition 4.10.

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