Towards the intersection theory on Hurwitz spaces

M. E. Kazaryan^{∗}, S. K. Lando ^{†}

February 25, 2004

Contents

1 Introduction  2
1.1 The Hurwitz problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Hurwitz spaces and the strata cohomology classes . . . . . . . . . . . 2
1.3 A description of main results . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 The degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Singularities, relative Chern classes, and universal polynomials …. 7
1.6 The structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Singularities of fibered mapswith one-dimensional fibers  10
2.1 Local singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Characteristic classes of singularities . . . . . . . . . . . . . . . . . . 11
2.3 Residual polynomials for multisingularities . . . . . . . . . . . . . . . 14
2.4 Residual polynomials for multimultisingularities . . . . . . . . . . . . 16
3 Relative Chern classes of the universal map  17
3.1 The Grothendieck–Riemann–Roch theorem . . . . . . . . . . . . . . . 17
3.1.1 The Chern character . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.2 The Todd class . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.3 The push-forward of a coherent sheaf . . . . . . . . . . . . . . 20
3.2 Relative Chern classes of the universal map . . . . . . . . . . . . . . . 20
3.2.1 The class c(q) and its pullback f^{∗}c(q) . . . . . . . . . . . . . . 22
3.2.2 The total relative Chern class of p . . . . . . . . . . . . . . . . 23
3.3 The pushforward homomorphisms . . . . . . . . . . . . . . . . . . . . 24
3.4 Relations for the cohomology classes in B . . . . . . . . . . . . . . . 25

4 Applying universal polynomials to the study of

∗Steklov Mathematical Institute RAS and the Independent University of Moscow †Institute for System Research RAS and the Independent University of Moscow, partly supported by the RFBR Grant 02-01-22004

stratification of Hurwitz spaces  25
4.1 Universal expressions for strata . . . . . . . . . . . . . . . . . . . . . 26
4.2 Degrees of the strata in genus 0 . . . . . . . . . . . . . . . . . . . . . 27
4.3 On nonisolated singularities . . . . . . . . . . . . . . . . . . . . . . . 29

1 Introduction

1.1 The Hurwitz problem

In paper [8] A. Hurwitz posed the following problem: count the number of isomorphism classes of branched coverings of the 2-sphere by a surface of genus g, having prescribed ramification points in the target sphere and prescribed ramification type over each ramification point. (We give a precise statement of the problem below.) During the last century, this problem attracted the researcher’s attention many times, however, this direction of investigations became a central one only recently. This explosion of interest is due, first of all, to the applications the Hurwitz problem found in modern mathematical physics, namely, in the string theory and in the theory of Gromov–Witten invariants.

Up to now, the Hurwitz problem does not have a satisfactory solution: the known formulas expressing the numbers of coverings in terms of the ramification data lead to transparent answers only in few partial cases. However, some hopes for finding explicit answers are revived due to the connection, found first in [1, 2], between the enumeration of ramified coverings of the sphere and the intersection theory on moduli spaces of meromorphic functions on algebraic curves. In these papers the Hurwitz problem is reduced, in fact, to a problem about the geometry of the moduli spaces. The latter is of its own interest. Because of the construction in [2], it is closely related to the now classical problem about the geometry of moduli spaces of algebraic curves.

The present paper is devoted to the analysis of the part of the geometry of Hurwitz spaces that can shed some light on the Hurwitz problem.

1.2 Hurwitz spaces and the strata cohomology classes

Consider ramified coverings of the 2-sphere by surfaces of genus g, and let n denote the degree of the coverings. Besides topological coverings we will also consider meromorphic functions of degree n on algebraic curves of genus g. Of course, each such function is a ramified covering from the topological point of view. Denote by H_{g,n} the space of such functions possessing the following properties:

• all the poles are simple; that is, a function has n poles of order one, and we assume that the poles are numbered;

• the sum of the critical values of a function is 0.

According to [2], this space is a smooth complex orbifold (or even a smooth variety, provided g = 0 or n is sufficiently large). It is fibered over the moduli space M_{g,n} of complex curves of genus g with n marked points: one can associate to each meromorphic function its curve of definition, with n marked points.

The space H_{g,n} admits a completion, denoted by \overline{H}_{g,n}, which consists of stable meromorphic functions [2, 14]. The boundary \overline{H}_{g,n} \ H_{g,n} consists of stable functions defined on singular curves, but the only singularities arising here are nodes (simple transversal intersections). The projection H_{g,n} → M_{g,n} extends to a projection \overline{H}_{g,n} → \overline{M}_{g,n} of the completed space to the moduli space of stable curves with marked points. The fibers of the latter projection are vector spaces since linear combinations of meromorphic functions having poles of order at most one at the marked points also possess the same property. Note, however, that generally speaking this projection is not a vector bundle, since the dimension of the fiber may vary from one point of the base to another. The fiberwise projectivization P \overline{H}_{g,n} of the completed Hurwitz space is a compact complex orbispace. It will serve as our main moduli space, while all other necessary spaces will be constructed by means of it. By some abuse of language, we will speak below about varieties and subvarieties having in mind that our moduli spaces are usually orbifolds and suborbifolds (that is, locally they look like quotients of a complex disc modulo a finite group action).

Being a projectivization, the space P\overline{H}_{g,n} carries a natural 2-cohomology class, namely, the first Chern class of the tautological sheaf, which we denote by ψ = ψ_{g,n} = c_{1}(O(1)) ∈ H^{2}(P \overline{H}_{g,n}).

On the other hand, this space contains subvarieties consisting of functions with degenerate ramifications. By the Riemann–Hurwitz formula, a general meromorphic function of degree n on a curve of genus g has 2n + 2g − 2 nondegenerate ramification points (both in the source and in the target). The functions having less ramification points in the target sphere form the discriminant in the space of functions. One can associate to each ramification point in the target a partition of n, which is the (unordered) set of multiplicities of the preimages of the given point. It will be more convenient for us, however, to make use of reduced partitions, that is, sets of multiplicities of the singular preimages of the degenerate ramification points, each diminished by 1. The closure in P \overline{H}_{g,n} of the set of functions having ramifications of prescribed types will be denoted by σ_{α_{1},...,α_{c}}, where the index consists of the reduced partitions over the degenerate ramification points. We call these subvarieties the strata of the discriminant. Thus, σ_{2^{1}} ⊂ P \overline{H}_{g,n} denotes the caustic, i.e., the stratum consisting of functions with two ramification points in the source glued together, while σ_{1^{2}} is the Maxwell stratum whose open part consists of functions taking the same value at two distinct critical points. The caustic and the Maxwell stratum are the only two strata of (complex) codimension one. Similarly, the stratum σ_{2^{1},1^{2}}, of codimension 2, is the closure of the set of functions having two degenerate critical values, with a ramification point of multiplicity three over one of them, and two ramification points of multiplicity 2 over the other one, and so on. The number of nondegenerate critical values (those corresponding to the reduced partition 1^{1}) for a generic function in a given stratum can be easily computed by means of the Riemann–Hurwitz formula. For the sake of notation simplicity, we do not include these nondegenerate reduced partitions, as well as the values g and n, in the notation for the strata.

Each stratum is a complex subvariety of pure dimension in P \overline{H}_{g,n} and determines, therefore, by the Poincaré duality, a homogeneous element of the cohomology ring H^{∗}(P \overline{H}_{g,n}) = H^{∗}(P \overline{H}_{g,n}, ℚ). (Everywhere below we study cohomology with rational coefficients; that is why we omit the reference to the ground ring in the notation for cohomology.) The degree of this element coincides with the (real) codimension of the stratum. Its product with the complementary degree of the class ψ is just a rational number, and — which is a result of the geometric approach suggested in [2]

— this number is closely related to the number of ramified coverings we are interested in. Before describing this relationship let us first give a precise formulation of the Hurwitz problem.

Two ramified coverings f_{1} : C_{1} → S^{2}, f_{2} : C_{2} → S^{2} of the 2-sphere are said to be isomorphic if there is a homeomorphism h : C_{1} → C_{2} such that f_{1} = f_{2} ∘ h. Clearly, isomorphic ramified coverings have the same ramification points in the target sphere, with the same ramification types over these points, and their degrees n_{1}, n_{2} and the genera g_{1}, g_{2} of the covering surfaces coincide, n_{1} = n_{2} = n, g_{1} = g_{2} = g. A ramification point in the target sphere is said to be nondegenerate, or simple, if it has n − 1 geometrically distinct preimages, among them one ramification point of order 2 and n − 2 smooth points of the covering. As it was already mentioned, we associate to each ramification point in the target sphere a reduced partition α, which consists of the multiplicities of its singular preimages, diminished by 1. The set of all partitions α, for a given ramified covering, form the ramification data. Denote by h_{α_{1},α_{2},...} the Hurwitz number, which equals, by definition,

h_{α_{1},α_{2},...} = ∑_{f} \frac{1}{|Aut(f)|},

where the summation is carried over all isomorphism classes f of ramified coverings of the sphere of degree n by the surface of genus g having reduced partitions α_{1}, α_{2}, … over the degenerate ramification points, and |Aut(f)| denotes the number of elements in the automorphism group of such a ramified covering (this group is always finite). Of course, the Hurwitz numbers also depend on g and n, but we do not include these parameters into the notation.

The Hurwitz problem consists in computing the Hurwitz numbers, and it is related to the geometry of the Hurwitz spaces by the following theorem, first stated for the case g = 0 in [16] and following immediately from [2].

Theorem 1.1 We have

h_{α_{1},α_{2},...} = \frac{|Aut(α_{1}, α_{2}, …)|}{n!}⟨σ_{α_{1},α_{2},...}ψ^{d}⟩,

where |Aut(α_{1}, α_{2}, …)| denotes the order of the automorphism group of the set of reduced partitions, that is, the product of factorials of the numbers of coinciding partitions, and the angle brackets denote the multiplication in the cohomology H^{∗}(P \overline{H}_{g,n}) of P \overline{H}_{g,n}.

This theorem reduces the Hurwitz problem to a question about the cohomology of Hurwitz spaces. In spite of the apparent complication of the problem caused by the replacement of a group combinatorial problem by an algebro-geometric one, the geometric interpretation not only clarifies the known computational results (see [8, 6, 15, 2]), but also leads to new ones (see, e.g. [16]).

Generally speaking, the cohomology rings of moduli spaces of stable curves and closely related Hurwitz spaces are very complicated, even in the case of rational functions (g = 0). Our experience shows, however, that all natural geometric classes related to the calculus of singularities (as well as multisingularitites and multimultisingularities, these are the classes playing the crucial role in the Hurwitz problem) can be expressed in terms of relatively small set of “basic” (tautological, in a sense) classes. Our aim is to identify these basic classes, to describe relations between them and to express the strata in terms of these classes.

1.3 A description of main results

Let us start with the results concerning the Hurwitz numbers.

Theorem 1.2 The following formulas take place for rational (i.e., referring to the case g = 0) Hurwitz numbers:

h_{2^{1},2^{1}} = \frac{3}{4} (27 n^{2} − 137 n + 180) \frac{(2 n − 6)!}{(n − 3)!} n^{n−6}
h_{2^{1},1^{2}} = 3 (3 n^{2} − 15 n + 20) \frac{(2 n − 6)!}{(n − 4)!} n^{n−6}
h_{1^{2},1^{2}} = 2 (2 n^{3} − 16 n^{2} + 43 n − 40) \frac{(2 n − 6)!}{(n − 4)!} n^{n−6}.

These formulas refer to multimultisingularities, that is, the situation where the ramification over several points in the target sphere is degenerate. For multisingularities, where the ramification is degenerate over a single point, the rational Hurwitz numbers are given by the well-known Hurwitz formula. The first formula in Theorem 1.2 is obtained by D. Zvonkine in [20], the two others, as far as we know, are new.

The proof of these results is based on some relations in the cohomology of Hurwitz spaces on classes represented by discriminant strata and the class ψ. We shall also make an extensive use of the class δ ∈ H^{2}(P \overline{H}_{g,n}) represented by functions defined on singular curves. The class δ is the pullback of the boundary class in \overline{M}_{g,n} under the projection \overline{H}_{g,n} → \overline{M}_{g,n}. Its importance in the geometry of the moduli spaces of stable curves is well-known. For example, the index of maximal self-intersection of this class is the Weil–Petersson volume of the moduli space, see [17]. The principal innovation produced in the study of Hurwitz spaces, is the presence of another natural 2-cohomology class, namely, the class ψ.

Note that all the results below are valid not only in the cohomology ring, but also in the Chow ring of P \overline{H}_{g,n}. The only reason for speaking about cohomology is that cohomology calculations are sufficient for the prospective computational applications.

As far as we know, the first result concerning the relations for strata in cohomology rings in Hurwitz spaces was obtained in [16]. It expresses the caustic and the Maxwell stratum in spaces of rational functions (g = 0) in terms of the boundary stratum and the class ψ:

\begin{aligned}σ_{2^{1}} & = & 6(n − 1)ψ − 3δ \\ σ_{1^{2}} & = & 2(n − 1)(n − 6)ψ + 4δ. \end{aligned}  (1)

Below we give this fact a new derivation. Classes of complex codimension greater than one also are subject to numerous relations, which are described in more detail in the main body of the paper since their formulation requires additional definitions.

1.4 The degree

Theorem 1.1 reduces the computation of a Hurwitz number to the calculation of the intersection index of a stratum with the class ψ, taken in the complementary degree. Such a class ψ naturally arises on any space of orbits of the group ℂ^{∗} action, and it behaves functorially under equivariant maps of varieties endowed with actions of this group. That is why the intersection of a subvariety with the complementary degree of this class deserves the name of “degree”.

Let A be a complex variety, and suppose the multiplicative group ℂ^{∗} of nonzero complex numbers acts on it without fixed points. Let B be the variety of orbits of the ℂ^{∗}-action on A and suppose B is compact. Denote by Ψ = c_{1}(O(1)) ∈ H^{2}(B) the tautological class of the ℂ^{∗}-action. For any class β ∈ H^{∗}(B) we call the degree of β the result of the pairing

deg β = ∫_{B} \frac{β}{1 − Ψ} = ∫_{B} β(1 + Ψ + Ψ^{2} + Ψ^{3} + …).

In other words, the degree of a homogeneous class β is its intersection index with the class Ψ taken in the degree equal to the codimension of β.

Example 1.3 A weighted projective space ℂP_{w_{0},...,w_{n}} is the quotient space of the complement to the origin in the vector space ℂ^{n+1} endowed with the following action of ℂ^{∗}:

λ : (x_{0}, …, x_{n}) ↦ (λ^{w_{0}}x_{0}, …, λ^{w_{n}}x_{n}), λ ∈ ℂ^{∗}.

The integer exponents w_{i} are called the weights of the action. Each linear representation of ℂ^{∗} has such a form. The degree of a weighted projective space ℂP_{w_{0},...,w_{n}} with weights w_{0}, …, w_{n} is (∏ w_{i})^{−1}.

The following functoriality property of the degree follows immediately from the definitions.

Proposition 1.4 Suppose f : B_{1} → B_{2} is a finite morphism of two compact orbispaces induced by an equivariant morphism g : A_{1} → A_{2} of two complex varieties of the same dimension. Then deg B_{1} = deg f · deg B_{2}.

Theorem 1.1 is an immediate consequence of this proposition if we consider the map L_{g,n} : P \overline{H}_{g,n} → ℂP_{2,3,...,2n+2g−2} taking a meromorphic function to the unordered set of its critical values (both the function and the set of its critical values are considered up to multiplication by a constant).

1.5 Singularities, relative Chern classes, and universal polynomials

Equations (1) are obtained in [16] by counting intersection indices of the classes σ_{2^{1}}, σ_{1^{2}} with a basic set of classes of complementary dimension (that is, of complex dimension 1, curves). However, as the genus or the codimension of the strata grow, this method becomes cumbersome: both the choice of a basic set of classes of complementary dimension and the computation of the intersection indices with such a class are complicated problems, without any effective solutions known. In the present paper, we suggest another approach to the search for relations among the cohomology classes. This approach is based on Thom’s theory of universal polynomials of singularities, extended to the case of multisingularities in [13, 12, 11, 10]. Let us explain briefly the basics of this theory.

Let F : M → N be a general holomorphic map of smooth complex varieties (we assume, for definiteness, that the varieties are of the same dimension: although the theory works for other cases as well, we will apply it only in the situation of coinciding dimensions). The total Chern class of F is the fraction

c(F) = \frac{c(F^{∗}T N)}{c(T M)} = 1 + c_{1}(F^{∗}(T N) − c_{1}(T M))
+(c_{2}(F^{∗}T N) − c_{1}(F^{∗}T N)c_{1}(T M) + c_{1}^{2}(T M)) + ⋯ ∈ H^{∗}(M);

here c(·), for a vector bundle argument, denotes, as usual, the total Chern class of the vector bundle. The homogeneous components of the total Chern class are called the Chern classes of F and are denoted by c_{i}(F) ∈ H^{2i}(M):

c_{1}(F) = c_{1}(F^{∗}T N) − c_{1}(T M)
c_{2}(F) = c_{2}(F^{∗}T N) − c_{1}(F^{∗}T N)c_{1}(T M) − c_{2}(T M) + c_{1}^{2}(T M)

and so on. A map F has a singularity at a given point if its differential dF is degenerate at this point. Singularities are classified according to their types. The closure of the locus of points where F has a singularity of a given type determines, by the Poincaré duality, a cohomology class in H^{∗}(M). Thom’s theory states that this class can be expressed as a polynomial in the Chern classes of F, and the coefficients of this polynomial depend only on the type of the singularity, but not on the varieties M and N or the map F.

The notion of both a general map and the type of a singularity should be defined more exactly, see, e.g. [13]. Here we give only the simplest example, that of the singularity A_{1}: a map has a singularity A_{1} at a given point if the rank of its differential at this point is one less than the maximal one and the restriction of the map to the kernel of the differential is nondegenerate. For a general map, the locus of points with the A_{1}-singularity is a subvariety of complex codimension 1. The cohomology class of its closure simply coincides with c_{1}(F) = c_{1}(F^{∗}T N) − c_{1}(T M), which can be easily seen by considering the pullback of a differential 1-form on N to M. Already in the simplest cases this equation for the class A_{1} produces nontrivial results. Suppose, for example, that the dimension of both varieties is 1, i.e., we deal with a ramified covering, say of degree n, of a complex curve by a complex curve. Then we obtain the Riemann–Hurwitz formula for the number of ramification points of the ramified covering:

c_{1}(F) = n(2 − 2g_{N}) − (2 − 2g_{M}) = 2(n − 1) − 2(ng_{N} − g_{M}),

where g_{M} and g_{N} are the genus of the covering and the covered surface, respectively.

A similar statement for multisingularities states that the closure in N of the locus of points such that F has singularities of prescribed types at these points is described as a universal polynomial in the pushforwards F_{∗}((c_{1}(F))^{k_{1}}(c_{2}(F))^{k_{2}} …) ∈ H^{∗}(N) of monomials in the Chern classes of F. We also extend this approach to the case of multimultisingularities, that is, subvarieties in moduli spaces of functions consisting of functions with prescribed multisingularities.

We apply the universal polynomials approach to the universal map over the Hurwitz space, which we are now going to define. Denote by \overline{U}_{g,n} the universal curve over \overline{H}_{g,n}, i.e., a complex orbifold together with a map \overline{U}_{g,n} → \overline{H}_{g,n} whose fiber over a point f ∈ \overline{H}_{g,n} is the curve of definition of f (or, to be more precise, the quotient of this curve modulo the action of the automorphism group of f). Another variety is simply the direct product ℂP^{1} × \overline{H}_{g,n}, with a distinguished coordinate along the first factor. The universal map over the Hurwitz space takes a point of the universal curve \overline{U}_{g,n} to the value of the corresponding function at this point. It is fibered over the Hurwitz space.

Remark 1.5 It is well known that the universal curve over the moduli space of stable curves \overline{M}_{g,n} coincides with the moduli space \overline{M}_{g,n+1} of stable curves with n + 1 marked points. Indeed, a point in a fiber of the universal curve can be considered as the (n + 1) st marked point along the fiber. Hurwitz spaces admit a similar construction. Namely, the space \overline{H}_{g,n+1} can be identified with the total space of the relative tangent bundle over the universal curve \overline{U}_{g,n}. Indeed, a point of a fiber determines the (n+1) st marked point, while a tangent vector at this point determines the principal part of a function with a pole of order one at this point.

The multiplicative group ℂ^{∗} of nonzero complex numbers acts on each of the three spaces \overline{H}_{g,n}, \overline{U}_{g,n}, and ℂP^{1} × \overline{H}_{g,n} by multiplying a function by a constant. On \overline{U}_{g,n}, this action is fiberwise trivial, while ℂ^{∗} acts as multiplication by a constant on a fiber ℂP^{1} of the direct product (the latter action is well defined since we have chosen a coordinate along the fiber). The universal map is equivariant with respect to these actions, and the strata in the Hurwitz space are invariant with respect to them. After restricting the universal curve and the direct product to the complement to the locus of fixed points of the action on \overline{H}_{g,n} and taking the quotient of the resulting spaces modulo the action we obtain the triangle of spaces and maps which will be the main subject of our study. Note that since the action of ℂ^{∗} on the fibers of ℂP^{1} × \overline{H}_{g,n} is nontrivial, the quotient of the direct product does not coincide with the direct product ℂP^{1} × P \overline{H}_{g,n}; instead, it is isomorphic to the projectivization of the rank two vector bundle ℂ ⊕ O(1) over P \overline{H}_{g,n}.

Introduce the simplified notation B_{g,n} for the projectivized Hurwitz space P \overline{H}_{g,n}, X_{g,n} for the quotient of the universal curve (which is, in its own turn, the universal curve over the projectivized Hurwitz space), Y_{g,n} for the quotient of the direct product, p_{g,n} (respectively, q_{g,n}) for the projection of X_{g,n} (respectively, Y_{g,n}) to B_{g,n}, and f_{g,n} for the factorized universal map, f_{g,n} : X_{g,n} → Y_{g,n}. In other words, we consider the commutative triangle

\begin{matrix}X_{g,n} & ⟶^{f_{g,n}} & Y_{g,n} \\ _{p_{g,n}} ↘ & ↙_{q_{g,n}} \\ B_{g,n} \end{matrix}  (2)

Our first goal is to show that the relative Chern classes of the universal map can be expressed in terms of few ‘basic’ classes in H^{∗}(X_{g,n}) and then describe the strata in the space B_{g,n} in terms of the pushforwards of polynomials in these basic classes. We are able to do this because the types of singularities of f_{g,n} are few in number and they are independent of g and n, the fact that leads to universal formulas.

Note that since the Hurwitz space B_{g,n} is a moduli space of stable meromorphic functions of degree n on curves of genus g, and X_{g,n} is the universal curve over B_{g,n}, any family of stable meromorphic functions of degree n on curves of genus g is induced from the projection p_{g,n} : X_{g,n} → B_{g,n}. In particular, linear relations among cohomology classes that are valid in X_{g,n} and B_{g,n} are preserved in arbitrary families.

1.6 The structure of the paper

In the next section we analyze local singularities that the universal map can acquire, and introduce basic cohomology classes. Then we show that the relative Chern classes of the universal map indeed can be expressed in terms of the basic classes. Next we produce expressions for some strata in B_{g,n} in terms of pushforwards of polynomials in the basic classes. The last section is devoted to calculations of the degrees of the strata.

1.7 Acknowledgements

We are grateful to D. Zagier and D. Zvonkine for useful discussions. The work on the paper has been completed during the second author’s stay at the Max-Planck-Institut für Mathematik, Bonn, in January–February 2004.

2 Singularities of fibered maps with one-dimensional fibers

The relative Chern classes of the universal map f_{g,n} are closely related to the singularities of this map. Since the map f_{g,n} is fibered over the base B_{g,n} and the fibers are one-dimensional, the singularities are not too complicated. The classification of simple isolated singularities of fibered maps with one-dimensional fibers [5] includes singularities of types A_{k} and I_{k,l}. The map f_{g,n} acquires singularities of these types. However, besides isolated singularities this map also has (with the exception of the simplest cases g = 0, n = 2, or n = 3) nonisolated ones. The latter arise if the underlying fiber possesses an irreducible component where the function is constant. Note that being constant on an irreducible component does not necessarily make the function unstable: it remains stable if either the genus of the component is at least 1, or, for a rational component, if it has at least three nodes. Of course, a complete study of Hurwitz spaces following our approach requires an analysis of nonisolated singularities as well, which we do not present here, and we obtain only partial results. We plan to discuss nonisolated singularities in a separate paper.

2.1 Local singularities

Consider the commutative triangular diagram of spaces and holomorphic maps of the form

\begin{matrix}ℂ^{m+1} & ⟶^{f} & ℂ × ℂ^{m} \\ _{p} ↘ & ↙_{q} \\ ℂ^{m} \end{matrix}  (3)

where q is the projection of the direct product to the second factor. We suppose that the fibers of p are one-dimensional (although not necessarily smooth). Restricting f to the fibers of p we obtain a family of functions on (noncompact) curves. The natural ‘right’ equivalence of such diagrams allows arbitrary coordinate changes in the base ℂ^{m}, as well as fiberwise coordinate changes in the source space ℂ^{m+1} fibered over the chosen coordinate change in the base. We add to this equivalence group the possibility to multiply the function by an arbitrary constant holomorphically depending on the point of the base. Such extension does not affect the classification of singularities, but leads to a more rich theory of characteristic classes.

Since the map q is a given projection, and p is the composition of f and q, the singularities of the diagram are totally determined by that of f. A point x ∈ ℂ^{m+1} is a local singularity of f if the rank of f at x is smaller than m + 1. We make the following two assumptions:

• all singularities f acquires are generic, i.e., the complement to the set of such singularities forms an open dense subset in the space of the diagram jets;

• the projection p : ℂ^{m+1} → ℂ^{m} has no singularities more complicated than the Morse fold (x, y) ↦ xy.

For the universal map we are interested in, the second assumption is satisfied, while the first one is not: the restriction of a map to a fiber may prove to be constant on some irreducible component of the fiber. As a result, the formulas we obtain are true for the universal mapping ‘up to some classes supported on the subvariety of nonisolated singularities’. However, in the case of rational functions the subvariety of nonisolated singularities has codimension 3, which allows us to obtain exact formulas for up to codimension 2 strata.

Under the above assumptions a map f has singularities of the following types:

• singularities of type A_{k}, k = 1, 2, …, which are realized at those smooth points of fibers of p where the restriction of f to the fiber has the form x ↦ x^{k+1} in a suitable coordinate x;

• singularities of type I_{k,l}, k ≥ l ≥ 1, which are realized at those singular points of fibers of p where the restriction of f to the plane with the coordinates x, y such that p looks like (x, y) ↦ xy in these coordinates has the form f : (x, y) ↦ (x^{k} + y^{l}, xy).

At these points, one can choose coordinates on the base ℂ^{m} such that f is a direct sum of an identity map and the following miniversal unfoldings, for A_{k} and I_{k,l}, respectively:

• A_{k}:(x, b_{2}, …, b_{k−1}) ↦ (x^{k+1} + b_{2}x^{k−1} + ⋯ + b_{k}x, b_{2}, …, b_{k})

• I_{k,l}: (x, y, a_{1}, …, a_{k−1}, b_{1}, …, b_{l−1}) ↦ (x^{k} + a_{1}x^{k−1} + ⋯ + a_{k−1}x + y^{l} + b_{1}y^{l−1}+ ⋯ + b_{l−1}y, xy, a_{1}, …, a_{k−1}, b_{1}, …, b_{l−1}).

The action of the group ℂ^{∗} in these coordinates looks as follows:

• for the singularity A_{k}:

λ : (x, b_{2}, …, b_{k}) ↦ (λx, λ^{2}b_{2}, …, λ^{k}b_{k}), λ ∈ ℂ^{∗};

• for the singularity I_{k,l}:

λ :(x, y, a_{1}, …, a_{k−1}, b_{1}, …, b_{l−1})
↦ (λ^{l}x, λ^{k}y, λ^{l}a_{1}, …, λ^{l(k−1)}a_{k−1}, λ^{k}b_{1}, …, λ^{k(l−1)}b_{l−1}), λ ∈ ℂ^{∗}.

2.2 Characteristic classes of singularities

Suppose that diagram (3) has a singularity of type A_{k} at the origin. Without loss of generality we may suppose that k = m + 1 and f is simply the miniversal unfolding of A_{k}, as above. The space ℂ^{k} at the upper left corner of the diagram contains subvarieties corresponding to all singularities A_{i} with i ≤ k. These subvarieties are invariant with respect to the action of ℂ^{∗}, therefore, to the singularity type A_{i} a subvariety [A_{i}] in the weighted projective space ℂP^{(k)} = ℂP_{1,2,...,k} obtained by factorization of ℂ^{k} modulo the action of ℂ^{∗} is associated, whence a cohomology class [A_{i}] ∈ H^{2i}(ℂP^{(k)}). Denote by Σ ∈ H^{2}(ℂP^{(k)}) the characteristic class of the singularity A_{1}, that is, the first Chern class of f, and denote by Ψ ∈ H^{2}(ℂP^{(k)}) the characteristic class c_{1}(O(1)).

Theorem 2.1 The characteristic classes of the singularities [A_{i}] in the unfoldings of the singularities A_{k} are given by the formula

[A_{i}] = P_{i}(Σ, Ψ) = Σ(2Σ − Ψ)(3Σ − 2Ψ) … (iΣ − (i − 1)Ψ).

In particular, the expressions for these classes in terms of Ψ, Σ are independent of the order k of the deformed singularity A_{k}.

Proof. The cohomology of the space ℂP^{(k)} are induced from the cohomology of the classifying space BG of the group G = ℂ^{∗}. This classifying space is ℂP^{∞}, and its cohomology ring is isomorphic to the ring of polynomials Z[t] under the natural choice of the generator t. Therefore, all elements of the cohomology ring H^{∗}(ℂP^{(k)}) can be expressed as polynomials in the variable t. In particular, Ψ = (k + 1)t, Σ = kt. The expression for [A_{i}] in terms of t must be

[A_{i}] = c_{i} · (k)_{i}t^{i}

for some constant c_{i}. Here the Pohgammer symbol (k)_{i} denotes the polynomial of degree i in k equal to

(k)_{i} = k(k − 1)(k − 2) … (k − i + 1).

Indeed, [A_{i}], being homogeneous of degree i, is proportional to t^{i} and the coefficient of proportionality is a polynomial in k of degree i which vanishes for all k < i. Now, in the case i = k we have [A_{i}] = i!t^{i}, which means that c_{i} = 1. The expression for A[i] in terms of Ψ and Σ is obtained by substituting the values t = Ψ−Σ, k = Σ/(Ψ−Σ). The theorem is proved.

The miniversal unfolding of the singularity I_{k,l} contains strata corresponding to singularities of type A_{i}, for i < k+l as well as strata corresponding to the singularities of type I_{i,j}, for i ≤ k, j ≤ l. These strata are also invariant with respect to the ℂ^{∗} action. After taking the quotient modulo this action we obtain the weighted projective space ℂP^{(k,l)} = ℂP_{l,l,2l,...,(k−1)l,k,k,2k,...,(l−1)k}, and all the cohomology classes considered below in this section belong to the cohomology of this space.

The singularities of types I_{i,j} have support on the subvariety Δ of singular points of the projection p. In the above local coordinates, this subvariety is given by the equation x = y = 0. Let us add to the classes Ψ, Σ ∈ H^{2}(ℂP^{(k,l)}) the class Δ ∈ H^{4}(ℂP^{(k,l)}). Consider also the normal bundle to Δ and denote by N the first Chern class of this normal bundle. The class Δ itself can be understood as the second Chern class of this normal bundle, and these two classes can be treated as (commuting) operators acting on Δ. Even less formally, we may treat N as the cohomology class N ∈ H^{2}(ℂP^{(k,l)}) (which makes sense only when applied to Δ). We are interested in expressing the classes of singularities of type A_{i} in the unfoldings of I_{k,l} in terms of the classes Σ, Ψ, Δ, N.

Theorem 2.2 The classes [A_{i}] in the unfoldings of the singularities I_{k,l} can be represented as weighted homogeneous polynomials of degree i in the variables Σ, Ψ, Δ, N in such a way that • these polynomials are universal, that is, they depend only on i and not on k

and l;

• the polynomial corresponding to the class [A_{i}] can be represented as the sum of two polynomials, the first one coinciding with the polynomial P_{i}(Σ, Ψ) defined above, the second one having support on the stratum Δ of double points;

• any polynomial having support on Δ can be expressed as a polynomial in Δ, N, and Ψ.

The last statement of the theorem follows from the obvious relation ΣΔ = ΨΔ. Denote the polynomial corresponding to the class [A_{i}] by R_{i}. It has the form R_{i} =

P_{i} + Q_{i−2}Δ, where the polynomial P_{i} is already known, and Q_{i−2} = Q_{i−2}(Δ, N, Ψ) is a weighted homogeneous polynomial of degree i − 2. We did not manage to find an explicit formula for these polynomials, but they can be computed using the indeterminate coefficients method. Let us show how this method works for small i.

Similarly to the situation above, the cohomology ring of the weighted projective space ℂP^{(k,l)} is induced from the cohomology ring of the classifying space BG = ℂP^{∞} of the group G = ℂ^{∗}. Therefore, each element of H^{∗}(ℂP^{(k,l)}) can be expressed as a polynomial in t. In particular, we have

Ψ = Σ = klt,  N = (k + l)t,  Δ = klt^{2}.

Let us find, using these values, the coefficients of the polynomials Q_{0} and Q_{1}. We know that Q_{0} = a for some constant a and that the stratum A_{2} in the unfolding of I_{1,1} is empty. Here k = l = 1 and we have Ψ = Σ = t, Δ = t^{2}, whence

R_{2} = Σ(2Σ − Ψ) + aΔ = t^{2} + at^{2} = 0.

Therefore, a = −1, i.e.,

R_{2} = Σ(2Σ − Ψ) − Δ.

Similarly, Q_{1} = aN + bΨ for some constants a and b. Using the fact that the stratum A_{3} is empty in the unfoldings of each of the two singularities I_{1,1} and I_{2,1}, we obtain the following system of two linear equations:

\{ \begin{matrix}1 & + & 2a & + & b & = & 0 \\ 8 & + & 6a & + & 4b & = & 0 \end{matrix}

(we divided both equations by t^{3} thus making use of their homogeneity), whence a = 2, b = −5, i.e.,

Q_{1} = 2N − 5Ψ.

Similar calculations give

Q_{2} = −(6N^{2} − 15NΨ + 15Ψ^{2} − 8Δ)
Q_{3} = 24N^{3} − 62N^{2}Ψ + 63NΨ^{2} − 35Ψ^{3} − 60NΔ + 84ΨΔ
Q_{4} = −(120N^{4} − 322N^{3}Ψ + 343N^{2}Ψ^{2} − 196NΨ^{3} + 70Ψ^{4}
− 432N^{2}Δ + 812NΨΔ − 469Ψ^{2}Δ + 180Δ^{2}).

Note that for A_{i}, the number of indeterminate coefficients is [i^{2}/4], where the square brackets denote the floor, and this is exactly the number of I_{k,l}-singularities not containing the stratum A_{i} in their miniversal unfoldings.

2.3 Residual polynomials for multisingularities

Fix a partition α = 1^{m_{1}}2^{m_{2}} … (here only finitely many exponents are nonzero). Consider the set of points y in the space ℂ × ℂ^{m} in the upper right vertex of the commutative triangle (3) possessing the following properties:

• the fiber p^{−1}(q(y)) of the projection p is nonsingular;

• the restriction of f to this fiber has, among the preimages of y, exactly m_{1} copies of the singularity A_{1}, exactly m_{2} copies of the singularity A_{2}, and so on. (Besides singularities of prescribed types, the point y can also have nonsingular preimages, whose number is determined by the degree of f.)

Denote the closure of this set by A_{α} ⊂ ℂ × ℂ^{m}.

Introduce the exponential generating function in infinitely many variables t_{1}, t_{2}, … having formal coefficients R_{α}, where α runs over nonzero partitions:

R(t_{1}, t_{2}, t_{3}, …) = ∑ R_{1^{m_{1}}2^{m_{2}}} \frac{t_{1}^{m_{1}}}{m_{1}!} \frac{t_{2}^{m_{2}}}{m_{2}!} ….

We are going to prove that there exist weighted homogeneous polynomials R_{α} in the classes Σ, Ψ, Δ, N such that after substituting them as the coefficients into the generating series R the cohomology classes of the strata A_{1^{k_{1}}2^{k_{2}}...} coincide with the coefficients of the monomials t_{1}^{k_{1}}t_{2}^{k_{2}} … in the exponent exp(f_{∗}R).

Here f_{∗} denotes the Gysin homomorphism, i.e., the pushforward homomorphism in the cohomology of the quotient spaces modulo the ℂ^{∗}-action. This map is only additive, not multiplicative, structure homomorphism. The polynomials R_{α} introduced in this way are called the residual polynomials for multisingularities. Of course, there is nothing strange in the fact that the polynomials R_{k^{1}} coincide with the polynomials R_{k} in the previous section (indeed, in this case the points under consideration have only one singular preimage).

In order to compute the residual polynomials we must know how the homomorphism f_{∗} acts on the cohomology classes we are interested in. Since the target space of f also is a weighted projective space, its cohomology ring also is induced from H^{∗}(BG) being thus a ring of polynomials in t. Therefore, f_{∗} looks like f_{∗} : t^{i} ↦ ct^{i} for some constant c. This constant cannot be other than the constant f_{∗}1, that is, the degree of f. If f is the unfolding of A_{k}, then its degree is k + 1.

We start with computing the polynomials R_{α} for the standard unfoldings of singularities of type A_{k} (that is, in the presence of only singularities of type A). These polynomials depend only on the classes Σ and Ψ. They can be computed as follows.

After substituting Σ = kt, Ψ = (k + 1)t into the generating function R we must obtain

exp((k + 1)R(t_{1}, …)) = 1 + (k + 1)_{2} \frac{t_{1}}{1!}t + ((k + 1)_{3} \frac{t_{2}}{1!} + (k + 1)_{4} \frac{t_{1}^{2}}{2!}) t^{2}
+ ((k + 1)_{4} \frac{t_{3}}{1!} + (k + 1)_{5} \frac{t_{1}}{1!} \frac{t_{2}}{2!} + (k + 1)_{6} \frac{t_{1}^{3}}{3!}) t^{3} + …  (4)

Indeed, the coefficient of \frac{t_{1}^{i_{1}}}{i_{1}!} \frac{t_{2}^{i_{2}}}{i_{2}!} … in this exponent must be t^{i_{1}+2i_{2}+3i_{3}+...} times a polynomial in k which vanishes for all k = −1, 0, 1, …, 2i_{1} + 3i_{2} + ⋯ − 2, since this is the codimension of the corresponding stratum.

Note that the right-hand side of Eq. (4) can be conveniently rewritten in the form^{1}

exp((k + 1)R(t_{1}, …)) = e^{τ_{1}τ+τ_{2}τ^{2}+τ_{3}τ^{3}+...}s^{k+1}|_{s=1}.

Here s is an auxiliary variable commuting with t_{i} and t, and we use the notation

τ = t\frac{d}{ds}, τ_{i} = t_{i} \frac{d}{ds}.

Now, the residual polynomials R_{α} are easily calculated by taking the logarithm of the right-hand side of Eq. (4), dividing it by (k + 1) and substituting t = Ψ − Σ, k = Σ/(Ψ − Σ). For small codimensions we thus obtain

R_{1^{2}} = −Σ(3Σ − Ψ)
R_{1^{1}2^{1}} = −6Σ(2Σ − Ψ)^{2}
R_{1^{3}} = \frac{4}{3}Σ(15Σ^{2} − 13ΣΨ + 3Ψ^{2})
R_{1^{1}3^{1}} = −4Σ(5Σ − 3Ψ)(3Σ − 2Ψ)(2Σ − Ψ)
R_{2^{2}} = −\frac{3}{2}Σ(2Σ − Ψ)(20Σ^{2} − 25ΣΨ + 8Ψ^{2})
R_{1^{2}2^{1}} = −\frac{3}{2}Σ(2Σ − Ψ)(15Σ^{2} − 17ΣΨ + 5Ψ^{2})
R_{1^{4}} = −2Σ(105Σ^{3} − 160Σ^{2}Ψ + 84ΣΨ^{2} − 15Ψ^{3}).

The Δ-part of the residual polynomials in the unfoldings of the singularities I_{k,l} can be computed recursively, using the indeterminate coefficients method as above.

Let us compute, for example, the polynomial R_{1^{2}}. Because of weighted homogeneity, it has the form

R_{1^{2}} = ΣΨ − 3Σ^{2} + aΔ

for some constant a (the Δ-free part is already known), while the stratum A_{1^{2}} can be expressed as the unknown coefficient of t_{1}^{2}/2! in exp(f_{∗}R):

[A_{1^{2}}] = \frac{1}{2}(f_{∗}R_{1^{1}})^{2} + f_{∗}R_{1^{2}}
= \frac{1}{2}(f_{∗}Σ)^{2} + f_{∗}(ΣΨ − 3Σ^{2} + aΔ).

1We are obliged for this remark to D. Zagier.

For the unfolding of I_{k,l}, the degree is k + l. Substituting the expressions for the classes Ψ, Δ, Σ in terms of the class t for the unfolding of the singularity I_{1,1} and using the fact that the stratum A_{1^{2}} in this unfolding is empty, we obtain the linear equation

\frac{1}{2} · 2^{2} + 2 − 6 + 2a = 0

which yields a = 1, i.e.,

R_{1^{2}} = ΣΨ − 3Σ^{2} + Δ = −Σ(3Σ − Ψ) + Δ.

Similar calculations for residual polynomials of higher codimensional multisingularities produce the following answers:

R_{1^{1}2^{1}} = −6Σ(2Σ − Ψ)^{2} + 6Δ(3Ψ − N_{1})
R_{1^{3}} = \frac{4}{3}Σ(15Σ^{2} − 13ΣΨ + 3Ψ^{2}) − \frac{4}{3}Δ(10Ψ − 3N)
R_{1^{1}3^{1}} = −4Σ(5Σ − 3Ψ)(3Σ − 2Ψ)(2Σ − Ψ)

+4Δ(20Ψ^{2} − 17NΨ + 6N^{2} − 8Δ)

R_{2^{2}} = −\frac{3}{2}Σ(2Σ − Ψ)(20Σ^{2} − 25ΣΨ + 8Ψ^{2})

+\frac{3}{2}Δ(25Ψ^{2} − 21NΨ + 8N^{2} − 12Δ)

R_{1^{2}2^{1}} = −\frac{3}{2}Σ(2Σ − Ψ)(15Σ^{2} − 17ΣΨ + 5Ψ^{2})

−12Δ(20Ψ^{2} − 15NΨ + 5N^{2} − 7Δ)

R_{1^{4}} = −2Σ(105Σ^{3} − 160Σ^{2}Ψ + 84ΣΨ^{2} − 15Ψ^{3})

+2Δ(70Ψ^{2} − 48NΨ + 15N^{2} − 21Δ).

2.4 Residual polynomials for multimultisingularities

Now we are interested in the subvarieties in the lower vertex of the triangle (3) consisting of points b such that the restriction of f to the fiber p^{−1}(b) has several degenerate multisingularities with prescribed reduced partitions α_{1}^{m_{1}}, …, α_{c}^{m_{c}} (and the necessary complementary number of nondegenerate singularities). Here the exponent marks the number of occurences of the corresponding partition. Denote by A_{α_{1}^{m_{1}},...,α_{c}^{mc}} ⊂ ℂ^{m} the closure of the set of these points.

Introduce the exponential generating function, in infinitely many indeterminates s_{1^{2}}, s_{2^{1}}, … indexed by partitions α, with formal coefficients T_{α_{1}^{m_{1}},...,α_{c}^{mc}} indexed by multipartitions:

ℜ(s_{1^{2}}, s_{2^{1}}, …) = ∑ T_{α_{1}^{m_{1}},...,α_{c}^{mc}} \frac{s_{α_{1}}^{m_{1}}}{m_{1}!} … \frac{s_{α_{c}}^{m_{c}}}{m_{c}!}.

Then the following statement takes place.

Theorem 2.3 After choosing for T_{α} the polynomials expressing the classes A_{α}, there exist weighted homogeneous polynomials R_{α_{1}^{m_{1}},...,α_{c}^{mc}}, for c ≥ 2, in the classes Σ, Ψ, Δ, N such that after substituting their Gysin pushforwards f_{∗}R_{α_{1},...,α_{c}} for T_{α_{1}^{m_{1}},...,α_{c}^{mc}} in the above generating function ℜ the cohomology classes of the strata A_{α_{1}^{m_{1}},...,α_{c}^{mc}} coincide

with the coefficients of the monomial \frac{s_{α_{1}}^{m_{1}}}{m_{1}!} … \frac{s_{αc}^{mc}}{m_{c}!} in the exponent exp(q_{∗}ℜ).

We formulate the last statement as a theorem, although it does not deserve this name yet, and it would be better termed a ‘conjecture’. However, numerous calculations confirming it, as well as the similarity of this statement to that in the cases of singularities and multisingularities, ensure its validity.

Once again, the polynomials R_{α_{1},...,α_{c}} can be found inductively using the indeterminate coefficients method. The computations make use of the fact that the map p is proper, when restricted to Σ, and all the cohomology classes we are interested in have support on Σ. Hence the action of the Gysin homomorphism p_{∗} reduces, when expressed in terms of the basic class t, to multiplication by the degree of p|_{Σ}, which is the number of critical points of f on a fiber, that is, k + 1 for the singularity A_{k} and k + l for the singularity I_{k,l}. Here are the results of sample calculations:

R_{2^{1},2^{1}} = −Σ(2Σ − Ψ)(5Σ − 3Ψ) + \frac{1}{2}Δ(14Ψ − 5N)
R_{2^{1},1^{2}} = 3Σ(2Σ − Ψ)(5Σ − 2Ψ) − 3Δ(7Ψ − 2N)
R_{1^{2},1^{2}} = −\frac{3}{4}Σ(30Σ^{2} − 21ΨΣ + 4Ψ^{2}) + \frac{1}{4}Δ(57Ψ − 14N).

3 Relative Chern classes of the universal map

In this section we are going to show that the relative Chern classes of the universal map indeed belong to the subring in H^{∗}(X_{g,n}) generated by the basic classes. This will confirm that the classes [A_{k}], as well as the residual polynomials for multisingularities and multimultisingularities also belong to this subring. The main tool in the computation of the relative Chern classes of f_{g,n} is the Grothendieck–Riemann– Roch theorem. It also produces nontrivial relations for the Gysin pushforwards of polynomials in the basic classes. We start with recalling this theorem in the specific situation we are interested in. Our exposition follows [3, 18].

3.1 The Grothendieck–Riemann–Roch theorem

The Grothendieck–Riemann–Roch (below, simply GRR) formula compares the Chern character of a coherent sheaf over a variety M with the Chern character of the image of this sheaf under a map P : M → A. We suppose that both M and A are compact and smooth.

In one of the forms (which is probably the most useful one) the GRR formula reads

ch(P_{!}β) = P_{∗}(ch(β)td(P)^{−1}).  (5)

Here β is a coherent sheaf on M, ch denotes the Chern character, and td is the Todd class of the morphism P.

Let us give definitions of the notions participating in the statement of the GRR formula.

3.1.1 The Chern character

The Chern character of the sheaf of sections of a vector bundle or, simply, the Chern character of a vector bundle, is defined as the sum of exponents of the Chern roots

ch(E) = exp(e_{1}) + ⋯ + exp(e_{r}),

where r is the rank of the vector bundle E. In particular, for a line bundle E we have ch(E) = exp(c_{1}(E)).

Being a symmetric function in the Chern roots, the Chern character of a vector bundle can be expressed in terms of its Chern classes:

ch(E) = r + c_{1} + \frac{1}{2}(c_{1}^{2} − 2c_{2}) + \frac{1}{6}(c_{1}^{3} − 3c_{1}c_{2} + 3c_{3})

+\frac{1}{24}(c_{1}^{4} − 4c_{1}^{2}c_{2} + 4c_{1}c_{3} + 2c_{2}^{2} − 4c_{4}) +

\frac{1}{120}(c_{1}^{5} − 5c_{1}^{3}c_{2} + 5c_{1}c_{2}^{2} + 5c_{1}^{2}c_{3} − 5c_{2}c_{3} − 5c_{1}c_{4} + 5c_{5}) + …,

where each c_{i} is the i th elementary symmetric function in the Chern roots. Due to the properties of Chern roots, for an exact sequence of vector bundles

0 ⟶ E′ ⟶ E ⟶ E′′ ⟶ 0,

we have the Whitney formula

ch(E) = ch(E′) + ch(E′′).

The Whitney formula allows one to extend the notion of Chern character to arbitrary coherent sheaves. A coherent sheaf on M is either the sheaf of sections of some vector bundle, or, more generally, a sheaf that can be inserted as a last nontrivial term into an exact sequence of sheaves all whose elements but the last one are sheaves of sections of vector bundles (such an exact sequence is called a resolvent for the given sheaf). The Chern character of a coherent sheaf is defined as the alternating sum of the Chern characters of the other terms of the exact sequence.

If Σ ⊂ M is a subvariety of codimension 1 and i : Σ ↪ M is the inclusion, then the corresponding resolvent simply looks like

0 ⟶ N_{Σ}^{∨} ⟶ O_{M} ⟶ i_{∗}(O_{Σ}) ⟶ 0,

where N_{Σ} is a line bundle over M such that Σ coincides with the zero locus of its section. In a tubular neighborhood of Σ the bundle N_{Σ} simply coincides with the natural extension of the normal bundle to Σ in M. Now the Whitney formula yields

ch(i_{∗}(O_{Σ})) = ch(O_{M}) − ch(N_{Σ}^{∨}) = 1 − exp(−Σ)
= \frac{(1 − exp(−Σ))}{Σ}Σ = (\frac{Σ}{1 − exp(−Σ)})^{−1} Σ,

where we are able to replace the operator of intersection with the first Chern class of the normal line bundle by Σ because, by assumption, Σ coincides with the zero locus of some holomorphic section of the normal bundle.

If Z is a subvariety of codimension 2, then the resolvent becomes more complicated. Namely, it has the form

0 ⟶ Λ^{2}N_{Z}^{∨} ⟶ N_{Z}^{∨} ⟶ O_{M} ⟶ i_{∗}(O_{Z}) ⟶ 0,  (6)

where Λ^{2} denotes the skew-symmetric square of the vector bundle. This sequence is written in the assumption that Z is the zero set of a globally defined section of some vector bundle N_{Z} over M. We shall see that the final expression for ch(i_{∗}(O_{Z})) is determined by the restriction of N_{Z} to Z, which is the normal bundle of Z. Moreover, one can show (see [3]) that the resulting expression is universal and is valid without the assumption that the normal bundle to Z extends to the whole M. The Whitney formula yields

ch(i_{∗}O_{Z}) = ch(Λ^{2}N_{Z}^{∨}) + ch(O_{M}) − ch(N_{Z}^{∨}).

The fictitious vector bundle N_{Z}^{∨} is of rank 2 and its total Chern class is c(N_{Z}^{∨}) = 1 − N_{1} + N_{2}, where N_{i} = c_{i}(N_{Z}), i = 1, 2. Hence, its Chern character is

ch(N_{Z}^{∨}) = 2 − N_{1} + \frac{1}{2}(N_{1}^{2} − 2N_{2}) + \frac{1}{6}(3N_{1}N_{2} − N_{1}^{3}) + \frac{1}{24}(N_{1}^{4} − 4N_{1}^{2}N_{2} + 2N_{2}^{2})

−\frac{1}{120}(N_{1}^{5} − 5N_{1}^{3}N_{2} + 5N_{1}N_{2}^{2}) + ….

The vector bundle Λ^{2}N_{Z}^{∨} is linear and

c(Λ^{2}N_{Z}^{∨}) = 1 − N_{1},

whence

ch(Λ^{2}N_{Z}^{∨}) = exp(−N_{1}).

Now, combining these results we obtain

ch(i_{∗}O_{Z}) = N_{2}− \frac{1}{2}N_{1}N_{2}+⋯ = (1 − \frac{1}{2}N_{1} + \frac{1}{12}(N_{1}^{2} − N_{2}) − \frac{1}{24}(N_{1}^{3} − N_{1}N_{2}) …) Z,

since both Z and N_{2} are identified with the class of the zero section of the normal bundle N_{Z}.

3.1.2 The Todd class

We start with defining the Todd class of a vector bundle, and then turn to the Todd class of a mapping. For a vector bundle E over M, its Todd class is another symmetric function in the Chern roots, namely,

td(E) = \frac{e_{1}}{1 − exp(−e_{1})} … \frac{e_{r}}{1 − exp(−e_{r})}
= 1 + \frac{1}{2}c_{1} + \frac{1}{12}(c_{1}^{2} + c_{2}) + \frac{1}{24}c_{1}c_{2} + \frac{1}{720}(−c_{1}^{4} + 4c_{1}^{2}c_{2} + 3c_{2}^{2} + c_{1}c_{3} − c_{4}) + ….

For an exact sequence of vector bundles like above, we have

td(E) = td(E′)td(E′′).

The following formula is true for the top Chern class of a vector bundle E of rank r:

∑_{i=0}^{r}(−1)^{i}ch(Λ^{i}E^{∨}) = c_{r}(E)td(E)^{−1}.

The Todd class of a variety M is the Todd class of its tangent bundle T M. The Todd class of a proper mapping F : M → N is the Todd class of the difference F^{∗}T N − T M, that is,

td(F) = td(F^{∗}T N − T M) = \frac{td F^{∗}T N}{td T M} .

3.1.3 The push-forward of a coherent sheaf

For a coherent sheaf E over M, the push-forward P_{!}(E) of E with respect to a map P : M → A is the most complicated object among those defined in the present section. Methods of computing it depend on the relative dimensions of A and M. In particular, if the dimension of A coincides with that of M, then the image P_{!}(E) coincides with the direct image P_{∗}(E). Recall that, by definition, the direct image P_{∗}E of a sheaf E over M with respect to a map P : M → A is the sheaf whose group of global sections over a sufficiently small open set U ⊂ A coincides with the restriction of E to P^{−1}(U). If dim M is dim A + 1 (this is exactly the situation to which we will apply the GRR theorem), then

P_{!}(E) = R^{0}(P_{∗}(E)) − R^{1}(P_{∗}(E)),

where R^{0}(P_{∗}(E)) = P_{∗}(E) is the direct image of E, and R^{1}(P_{∗}(E)) is the first Grothendieck higher direct image. The latter can be computed by applying the Serre duality:

R^{1}(P_{∗}(E)) = (R^{0}(P_{∗}(E^{∨} ⊗ ω)))^{∨} = (P_{∗}(E^{∨} ⊗ ω))^{∨},

where ω is the relative dualizing sheaf.

3.2 Relative Chern classes of the universal map

Now let us return to the commutative triangle (2) of spaces and maps. From now on, we simplify the notation by omitting the indices g, n, i.e., X_{g,n} is denoted by X, f_{g,n} is denoted by f, and so on. Introduce the following cohomology classes in the space X, the universal curve over the projectivized Hurwitz space, which we call basic classes. Denote by ω the relative dualizing sheaf of the map p : X → B, that is, the only holomorphic line bundle over X whose fiber at a smooth point of a fiber of p coincides with the cotangent line to the fiber. The first Chern class c_{1}(ω) is an element of the second cohomology group H^{2}(X). Let Π denote the divisor of poles of the universal map f_{g,n}, that is, the set of points in X such that the restriction of f to the fiber of p has a pole at this point. Recall that, being a quotient modulo an action of the group ℂ^{∗}, each of the spaces X, Y and B carries the cohomology class c_{1}(O(1)) ∈ H^{2}, which we denote by Ψ_{X}, Ψ_{Y}, and ψ, respectively. We also introduce notation Σ = c_{1}(ω) + Ψ_{X} + 2Π for the first relative Chern class of f.

Besides the subvarieties and classes of codimension one, we will also make use of the class Δ ⊂ X represented by the set of singular points of the fibers of p. The normal bundle to Δ in X has rank two. Locally it admits a splitting into a direct sum of two line bundles. Indeed, each point of Δ is a point of transversal intersection of the two sheets of the fiber curve at this point, and the tangent lines to the sheets provide a splitting of the normal bundle. (In the case of rational curves, g = 0, this splitting can be extended to a global splitting, since each double point cuts a rational curve into two connected components, but we will not use this property below.) Denote by N the first Chern class of the normal bundle to Δ ⊂ X. This Chern class can be understood as an operator that can be applied to Δ. Its powers N^{i} also are such operators. The second Chern class of the normal bundle can be simply identified with Δ itself.

Lemma 3.1 The basic classes satisfy the following relations (in H^{∗}(X)):

Π^{2} = −Ψ_{X} Π, Σ Π = 0, Π Δ = 0, Σ Δ = Ψ_{X} Δ.

Proof. The equality Π Δ = 0 means that the poles of any function in the Hurwitz space belong to the smooth part of the curve so that the varieties Δ and Π do not intersect. Similarly, the equality Σ Π = 0 is the algebraic expression of the fact that all the poles are simple and there is no ramification over infinity.

Now, the dualizing sheaf ω is locally trivial over Δ and it is isomorphic to the conormal line bundle of the hypersurface Π at the points of this hypersurface. This implies the equalities c_{1}(ω) Δ = 0 and Π^{2} = −c_{1}(ω) Π. Substituting c_{1}(ω) = Σ − Ψ_{X} − 2Π we get

0 = (Σ − Ψ_{X} − 2Π) Δ = Σ Δ − Ψ_{X} Δ

and

Π^{2} = −(Σ − Ψ_{X} − 2Π) Π = Ψ_{X} Π + 2Π^{2},

respectively. This proves the equalities Σ Δ = Ψ_{X} Δ and Π^{2} = −Ψ_{X} Π of the lemma. Note that the last one is also the pullback of the equality Π_{Y}^{2} = −Ψ_{Y} Π_{Y} which takes place in Y, see the next section.

Corollary 3.2 Every polynomial in the basic classes can be represented in the form

P_{1}(Ψ_{X}) Π + P_{2}(Ψ_{X}, Σ) + P_{3}(Ψ_{X}, N, Δ) Δ,

where P_{1}, P_{2}, and P_{3} are certain polynomials.

The main goal of the present section is to prove the following statement and to deduce from it some relations on the pushforwards of the basic classes.

Theorem 3.3 The total relative Chern class of f is

c(f) = 1 + \frac{Σ}{1 − Σ + Ψ_{X}} − \frac{Δ}{1 + N + Δ}.  (7)

As an immediate corollary we obtain the following statement.

Corollary 3.4 If the only singularities of the map f, when restricted to the fibers of p, are of type A_{k} or I_{k,l}, then the residual polynomials R_{α_{1},...,α_{c}} for multimultisingularities belong to the subring in H^{∗}(X) generated by the basic classes Σ, Ψ_{X}, Δ, N.

Since p is the composition of the two maps f and q, p = f ∘ q, its total relative Chern class is the ratio of the total relative Chern class of p and the pullback of the total relative Chern class of q:

c(f) = \frac{f^{∗}c(q)}{c(p)}.

Therefore, in order to compute it it suffices to compute the two classes c(p), c(q) as well as the pullback of the latter class to X. This is exactly what we are going to do now.

3.2.1 The class c(q) and its pullback f^{∗}c(q)

The total space Y of the fibration q is the projectivization of the rank two bundle E = ℂ ⊕ O(1) over B. Denote by T the tautological line bundle over Y and set Π_{Y} = c_{1}(T^{∨}) = −c_{1}(T ). This class is subject to the relation c_{2}(q^{∗}(E)/T ) = 0, that is

Π_{Y}^{2} + Π_{Y} Ψ_{Y} = 0,  (8)

where Ψ_{Y} = q^{∗}(ψ) = c_{1}(O(1)) is the corresponding characteristic class of the ℂ^{∗}- action. This relation determines the cohomology ring of Y as an algebra over H^{∗}(B) via q^{∗}.

The lines ℂ⊕{0} and {0}⊕O(1) in the fibers of E correspond to the points ‘zero’ and ‘infinity’ in the fibers of q, respectively. These points form hypersurfaces with the Poincaré dual classes

c_{1}(Hom(T , O(1))) = Π_{Y} + Ψ_{Y} , c_{1}(Hom(T , ℂ)) = Π_{Y} ,

respectively. Thus, Π_{Y} can be regarded as the class dual to the ‘infinity locus’ in Y, and Eq. (8) geometrically means that the zero and infinity loci have no common points.

The total Chern class of q is inverse to the total Chern class of the relative tangent bundle ker(q_{∗}). This bundle is determined by the isomorphism ker(q_{∗}) ≃ Hom(T , q^{∗}(E)/T ). Therefore,

c(q) = \frac{1}{c(Hom(T , q^{∗}(E)/T ))} = \frac{1}{1 + Ψ_{Y} + 2Π_{Y}} .

Applying f^{∗} we get f^{∗}(Ψ_{Y}) = Ψ_{X} and f^{∗}(Π_{Y}) = Π. Therefore, we get finally

f^{∗}(c(q)) = \frac{1}{1 + Ψ_{X} + 2Π}.

3.2.2 The total relative Chern class of p

For the map p, the computation of the relative Chern class is slightly more complicated. The relative Chern class of p can be computed via the exact sequence

0 ⟶ p^{∗}T^{∨}B ⟶^{p^{∨}} T^{∨}X ⟶ ω ⟶ O_{Δ} ⟶ 0.  (9)

where, recall, ω denotes the relative dualizing sheaf of p. The subvariety Δ ⊂ X is exactly the locus of points, where the map p^{∗} fails to be of corank 1 (surjective) and is of corank 2. The structure sheaf O_{Δ} of Δ measures the difference between the relative dualizing sheaf ω and the sheaf of relative differentials.

Applying to the above exact sequence the Whitney formula we obtain

c(p^{∨}) = 1 − c_{1}(p) + c_{2}(p) − ⋯ = c(p^{∗}T^{∨}B − T^{∨}X) = \frac{c(O_{Δ})}{c(ω)} = \frac{c(O_{Δ})}{1 + c_{1}(ω)}.

In order to compute the Chern class c(O_{Δ})) we make use of the Coszul projective resolvent (6). Another application of the Whitney formula yields

c(O_{Δ}) = \frac{1 − N}{1 − N + Δ} = 1 − \frac{Δ}{1 − N + Δ}.

Another way to obtain the same formula is to use the GRR-formula for the embedding Δ ↪ X. Combining with the above expression for c(p^{∨}) we get

c(p) = \frac{1}{1 − c_{1}(ω)} (1 − \frac{Δ}{1 + N + Δ}) .

Now we are able to compute the total Chern class of f. Applying relations of Lemma 3.1, we get

c(f) = \frac{c(p)}{f^{∗}(c(q))} = \frac{1 + Ψ_{X} + 2Π}{1 − c_{1}(ω)} (1 − \frac{Δ}{1 + N + Δ})

= \frac{1 + Ψ_{X} + 2Π}{1 − Σ + Ψ_{X} + 2Π} − \frac{1 + Ψ_{X}}{1 + N + Δ} Δ

= (\frac{1 + Ψ_{X}}{1 − Σ + Ψ_{X}} − \frac{1 + Ψ_{X}}{1 + N + Δ} Δ) + (\frac{1 + Ψ_{X} + 2Π}{1 − Σ + Ψ_{X} + 2Π} − \frac{1 + Ψ_{X}}{1 − Σ + Ψ_{X}} )

The first summand is equal to the expression in the formula of Theorem 7; the second summand, after reducing to the common denominator, attains the numerator

(1 + Ψ_{X} + 2Π)(1 − Σ + Ψ_{X}) − (1 + Ψ_{X})(1 − Σ + Ψ_{X} + 2Π) = −2Π Σ = 0.

Theorem 7 is proved.

3.3 The pushforward homomorphisms

In order to apply the universal formulas for the classes of multisingularities, we need also to compute the pushforward homomorphisms f_{∗}, p_{∗}, q_{∗}. The following theorem asserts that the homomorphism p_{∗} determines the two others.

Theorem 3.5 For any collection of cohomology classes h_{1}, …, h_{s} on X satisfying the condition h_{i} Π = 0, i = 1, …, s we have

q_{∗}(f_{∗}(h_{1}) … f_{∗}(h_{s})) = ψ^{s−1}p_{∗}(h_{1}) … p_{∗}(h_{s}).

The condition h_{i}Π = 0 geometrically means that h_{i} can be represented by a cycle having no common points with the divisor Π of poles. This condition is satisfied for residual polynomials of multisingularities. Proof. Since q is the projectivization of a vector bundle, the direct image f_{∗}(h) of an arbitrary class h ∈ H^{∗}(X) can be represented in the form

f_{∗}(h) = q^{∗}(a) + q^{∗}(b) Π_{Y}

with certain classes a and b on B. The class b can be obtained by applying q_{∗} to both sides of this equation. From the equation q_{∗}Π_{Y} = 1 and the projection formula we have

b = q_{∗}(f_{∗}(h)) = p_{∗}(h).

The class a is obtained by a similar argument. Namely, multiplying both sides of the above equation by Π_{Y} + Ψ_{Y}, we get

f_{∗}((Ψ_{X} + Π) h) = q^{∗}(a) (Ψ_{Y} + Π_{Y}) + q^{∗}(b) (Ψ_{Y} + Π_{Y}) Π_{Y} = q^{∗}(ψ a) + q^{∗}(a) Π_{Y} ,

and applying q_{∗} we find a = p_{∗}((Ψ_{X} + Π) h). If the class h satisfies the assumption h Π = 0 of the theorem, then the answer simplifies:

f_{∗}(h) = q^{∗}(p_{∗}(a)) (Ψ_{Y} + Π_{Y}).

Therefore, under the assumptions of the theorem we have

f_{∗}(h_{1}) … f_{∗}(h_{s}) = (Ψ_{Y} + Π_{Y})^{s} ∏_{i=1}^{s} q^{∗}(p_{∗}(h_{i})) = (Ψ_{Y}^{s} + Ψ_{Y}^{s−1}Π_{Y}) ∏_{i=1}^{s} q^{∗}(p_{∗}(h_{i})).

Applying q_{∗}, we get from the projection formula the desired relation.

Thus to complete the computation of the cohomology classes of strata in the Hurwitz space P \overline{H}_{g,n} we should be able to compute the pushforward class p_{∗}(R) for any residual polynomial represented in the form R = P_{1}(Ψ_{X}, Σ) + P_{2}(Ψ_{X}, N, Δ) Δ. Set

ξ_{k} = p_{∗}(Σ^{k+1}), δ_{k,ℓ} = p_{∗}(N^{k}Δ^{ℓ+1}).

There are no universal relations between the classes ξ_{k} and δ_{k,ℓ} which would be valid for any g and n. Of course, for any particular g and n the cohomology space H^{∗}(P \overline{H}_{g,n}) is of finite dimension, so that for fixed g and n such relations do exist. Some of them are discussed in the next section.

3.4 Relations for the cohomology classes in B

Let us apply the GRR formula to the projection p : X → B and to the relative dualizing sheaf ω. Our approach follows that of [18], where the same thing is done for the universal curve projection onto the moduli space of curves. We have

ch(p_{!}ω) = p_{∗}(ch(ω) · td(p)^{−1}).

The sheaf on the left-hand side p_{!}ω = R^{0}p_{∗}ω − R^{1}p_{∗}ω is expressed in terms of the Hodge bundle Λ = Λ_{g,n} over B = P \overline{H}_{g,n} (recall that the fiber of the Hodge bundle at a point (C; f) is the space of holomorphic 1-forms over C, so that the rank of the bundle is g): R^{0}p_{∗}ω = p_{∗}ω = Λ and, by the Serre duality, R^{1}p_{∗}ω = p_{∗}ℂ = ℂ. The classes entering the right-hand side of the formula were computed in the previous sections: using the equality c_{1}(ω)Δ = 0, we have

ch(ω)td^{−1}(p) = ch(ω)td(ω^{∨} − O_{Δ}^{∨}) = e^{c_{1}(ω)} \frac{−c_{1}(ω)}{1 − e^{c_{1}(ω)}} td(−O_{Δ}^{∨})

= \frac{c_{1}(ω)}{1 − e^{−c_{1}(ω)}} + td(−O_{Δ}^{∨}),

where the Todd class of the (fictitious) sheaf −O_{Δ}^{∨} is determined by its Chern class

c(−O_{Δ}^{∨}) = \frac{1 + N + Δ}{1 + N} = 1 + \frac{Δ}{1 + N} .

Substituting to the GRR formula, we get

(g − 1) + ch_{1}(Λ) + ch_{2}(Λ) + ch_{3}(Λ) + ch_{4}(Λ) + ch_{5}(Λ) + …

= \frac{1}{2}p_{∗}(c_{1}(ω)) + \frac{1}{12}p_{∗}(c_{1}^{2}(ω) + Δ) − \frac{1}{720}p_{∗}(c_{1}^{4}(ω) + (N^{2} − 3Δ)Δ)

+ \frac{1}{30240}p_{∗}(c_{1}^{6}(ω) + (N^{4} − 5N^{2}Δ + 5Δ^{2})Δ) + ….

(10)

This relation says that the homogeneous components of the Chern character (and hence the Chern classes) of the Hodge bundle are determined by the pushforwards of the basic classes.

Note that the right-hand side of (10) has no terms of even positive degree. It follows that the even part of ch(Λ) is trivial, that is c(Λ + Λ^{∨}) = 1. In the case g = 0 the Hodge bundle is trivial so that the left-hand side of (10) is trivial in positive degrees. It provides non-trivial identities between the pushforwards of the basic classes:

p_{∗}(c_{1}^{2}(ω) + Δ) = 0, p_{∗}(c_{1}^{4}(ω) + (N^{2} − 3Δ)Δ) = 0, ….

4 Applying universal polynomials to the study of stratification of Hurwitz spaces

The knowledge of the residual polynomials for singularities, multisingularities, and multimultisingularities allows one to obtain cohomological expressions for the strata in the Hurwitz spaces. Unfortunately only complete formulas, including the contribution of nonisolated singularities, are required. That’s why we present universal expressions for strata and compute the degrees of the corresponding strata only in the case of rational functions and only for small codimensions.

4.1 Universal expressions for strata

In the table below we present the expressions for the strata of codimension one and two moduli classes supported on the strata of nonisolated singularities via pushforwards of the basic classes, from the computed residual classes. In this table, we use notation

ξ_{k} = p_{∗}(Σ^{k+1}), ξ_{0} = 2n − 2 + 2g, δ_{k,l} = p_{∗}(N^{k}Δ^{l+1}).

σ_{2^{1}} = −ψ ξ_{0} + 2 ξ_{1} − δ_{0,0}
σ_{1^{2}} = \frac{1}{2} ψ ξ_{0} (ξ_{0} + 2) − 3 ξ_{1} + δ_{0,0}
σ_{3^{1}} = 2 ξ_{0} ψ^{2} − 7 ξ_{1} ψ − 5 δ_{0,0} ψ + 6 ξ_{2} + 2 δ_{1,0}
σ_{1^{1}2^{1}} = − ξ_{0} (ξ_{0} + 6) ψ^{2} + 2 (ξ_{0} + 12) ξ_{1} ψ + (18 − ξ_{0}) δ_{0,0} ψ − 24 ξ_{2} − 6 δ_{1,0}
σ_{1^{3}} = \frac{1}{6} ξ_{0} (ξ_{0}^{2} + 6 ξ_{0} + 24) ψ^{2}

− \frac{1}{3} (9 ξ_{0} + 52) ξ_{1} ψ + \frac{1}{3} (3 ξ_{0} − 40) δ_{0,0} ψ + 20 ξ_{2} + 4 δ_{1,0}

σ_{2^{1},2^{1}} = \frac{1}{2} (ξ_{0} − 6) ξ_{0} ψ^{2} − (2 ξ_{0} − 11) ξ_{1} ψ + \frac{1}{2} δ_{0,0}^{2} + 2 ξ_{1}^{2} − 10 ξ_{2}

+ (ξ_{0} + 7) δ_{0,0} ψ − 2 ξ_{1} δ_{0,0} − \frac{5}{2} δ_{1,0}

σ_{2^{1},1^{2}} = −\frac{1}{2} ξ_{0} (ξ_{0}^{2} − 2 ξ_{0} − 12) ψ^{2} + (ξ_{0}^{2} + ξ_{0} − 27) ξ_{1} ψ − δ_{0,0}^{2} − 6 ξ_{1}^{2} + 30 ξ_{2}

− \frac{1}{2} (ξ_{0}^{2} + 42)δ_{0,0} ψ + 5 ξ_{1} δ_{0,0} + 6 δ_{1,0}

σ_{1^{2},1^{2}} = \frac{1}{8} (ξ_{0} − 4) ξ_{0} (ξ_{0}^{2} + 4 ξ_{0} + 6) ψ^{2} − \frac{3}{4} (2 ξ_{0}^{2} − 4 ξ_{0} − 21) ξ_{1} ψ + \frac{1}{2} δ_{0,0}^{2}

+ \frac{9}{2} ξ_{1}^{2} − \frac{45}{2} ξ_{2} + \frac{1}{4} (2 ξ_{0}^{2} − 4 ξ_{0} + 57) δ_{0,0} ψ − 3 ξ_{1} δ_{0,0} − \frac{7}{2} δ_{1,0}.

In the next table we present simplified expressions for the same strata in the case g = 0 with the identities

ξ_{0} = 2n − 2, ξ_{1} = 4 (n − 1) ψ − δ_{0,0}

of Sec. 3.4 taken into account. Note that in the case g = 0 the nonisolated singularities have codimension 3 in B, therefore, they give no contribution to the strata of the table.

σ_{2^{1}} = 6 (n − 1) ψ − 3 δ_{0,0}
σ_{1^{2}} = 2 (n − 6) (n − 1) ψ + 4 δ_{0,0}
σ_{3^{1}} = −24 (n − 1) ψ^{2} + 2 δ_{0,0} ψ + 6 ξ_{2} + 2 δ_{1,0}
σ_{1^{1}2^{1}} = 12 (n − 1) (n + 6) ψ^{2} − 6 n δ_{0,0} ψ − 24 ξ_{2} − 6 δ_{1,0}
σ_{1^{3}} = \frac{4}{3} (n − 1) (n^{2} − 17 n − 30) ψ^{2} + 4 (2 n − 1) δ_{0,0} ψ + 20 ξ_{2} + 4 δ_{1,0}
σ_{2^{1},2^{1}} = 2 (n − 1) (9 n + 10) ψ^{2} − 2 (9 n − 7) δ_{0,0} ψ + \frac{9}{2} δ_{0,0}^{2} − 10 ξ_{2} − \frac{5}{2} δ_{1,0}
σ_{2^{1},1^{2}} = 12 (n − 9) (n − 1) n ψ^{2} − 6 (n^{2} − 13 n + 11) δ_{0,0} ψ − 12 δ_{0,0}^{2} + 30 ξ_{2} + 6 δ_{1,0}
σ_{1^{2},1^{2}} = (n − 1) (2 n^{3} − 30 n^{2} + 145 n − 60) ψ^{2} + \frac{1}{2} (16 n^{2} − 144 n + 125) δ_{0,0} ψ

+ 8 δ_{0,0}^{2} − \frac{45}{2} ξ_{2} − \frac{7}{2} δ_{1,0}.

4.2 Degrees of the strata in genus 0

In the case g = 0, the compactified Hurwitz space P \overline{H}_{0,n} is fibred over the compactified moduli space \overline{M}_{0,n} of rational curves with marked points. The fiber is the projective space P E, where E = L_{1}^{∨}⊕⋯⊕L_{n}^{∨} is the Whitney sum of the tangent lines to the curve at the marked points. It follows that the cohomology of \overline{M}_{0,n} considered as an algebra over H^{∗}(\overline{M}_{0,n}) is generated by the class ψ = c_{1}(O(1)) which is subject to the relation

ψ^{n} + c_{1}(E) ψ^{n−1} + ⋯ + c_{n}(E) = 0

(cf. [16]). In particular, any class α in H^{2d}(P \overline{H}_{0,n}) can be represented in the form

α = π^{∗}(η_{d}) + π^{∗}(η_{d−1})ψ + π^{∗}(η_{d−2})ψ^{2} + …,

where π : P \overline{H}_{0,n} → \overline{M}_{0,n} is the natural projection and the classes η_{i} are defined on \overline{M}_{0,n}. It follows from the well-known relation π_{∗}ψ^{s} = c_{s−n+1}(−E) (see, e.g., [3]) that the degree of any class represented in such form can be computed from the relation

degα = ∫_{P \overline{H}_{0,n}} \frac{π^{∗}(η_{d}) + π^{∗}(η_{d−1})ψ + …}{1 − ψ} = ∫_{\overline{M}_{0,n}} \frac{η_{d} + η_{d−1} + η_{d−2} + …}{c(E)}.

Thus the computation of the degree is reduced to the computation of certain integrals over the moduli spaces of rational curves with marked points. The machinery of computing such integrals is well developed. Below we present the results of computation of the degrees of the pushforwards of the basic classes of codimension at most two:

deg(1) = n^{n−3}
deg(δ_{0,0}) = \frac{1}{2} (n − 1) (n + 6) n^{n−4}
deg(ξ_{2}) = \frac{1}{3} (n − 1) (17 n^{2} − 28 n + 12) n^{n−5}
deg(δ_{1,0}) = −\frac{1}{6} (n − 1) (n^{2} + 10 n − 120) n^{n−5}
deg(δ_{0,0}^{2}) = \frac{1}{12} (n − 1) (3 n^{3} + 31 n^{2} + 82 n − 120) n^{n−5}.

Substituting these equalities to the computed expressions for the singularity strata we obtain the following formulas for the degrees of the these strata:

deg(σ_{2^{1}}) = \frac{9}{2} (n − 2) (n − 1) n^{n−4}
deg(σ_{1^{2}}) = 2 (n − 3) (n − 2) (n − 1) n^{n−4}
deg(σ_{3^{1}}) = \frac{32}{3} (n − 3) (n − 2) (n − 1) n^{n−5}
deg(σ_{1^{1}2^{1}}) = 9 (n − 4) (n − 3) (n − 2) (n − 1) n^{n−5}
deg(σ_{1^{3}}) = \frac{4}{3} (n − 5) (n − 4) (n − 3) (n − 2) (n − 1) n^{n−5}
deg(σ_{2^{1},2^{1}}) = \frac{3}{8} (n − 2) (n − 1) (27 n^{2} − 137 n + 180) n^{n−5}
deg(σ_{2^{1},1^{2}}) = 3 (n − 3) (n − 2) (n − 1) (3 n^{2} − 15 n + 20) n^{n−5}
deg(σ_{1^{2},1^{2}}) = (n − 3) (n − 2) (n − 1) (2 n^{3} − 16 n^{2} + 43 n − 40) n^{n−5}.

By Theorem 1.1, this leads to the following formulas for the Hurwitz numbers:

h_{2^{1}} = \frac{9}{2} \frac{(2 n − 4)!}{(n − 3)!} n^{n−5}
h_{1^{2}} = 2 \frac{(2 n − 4)!}{(n − 4)!} n^{n−5}
h_{3^{1}} = \frac{32}{3} \frac{(2 n − 5)!}{(n − 4)!} n^{n−6}
h_{1^{1}2^{1}} = 9 \frac{(2 n − 5)!}{(n − 5)!} n^{n−6}
h_{1^{3}} = \frac{4}{3} \frac{(2 n − 5)!}{(n − 6)!} n^{n−6}
h_{2^{1},2^{1}} = \frac{3}{4} (27 n^{2} − 137 n + 180) \frac{(2 n − 6)!}{(n − 3)!} n^{n−6}
h_{2^{1},1^{2}} = 3 (3 n^{2} − 15 n + 20) \frac{(2 n − 6)!}{(n − 4)!} n^{n−6}
h_{1^{2},1^{2}} = 2 (2 n^{3} − 16 n^{2} + 43 n − 40) \frac{(2 n − 6)!}{(n − 4)!} n^{n−6}

of which those corresponding to a single degeneration are known by the Hurwitz formula.

The alternative approach to the compution of the degrees of the strata consists in a direct application of the Hurwitz formula (or, for arbitrary genus, of the formula in [1], which is, however, less explicit) for the Hurwitz numbers of multisingularities. Via Theorem 1.1, it gives us the degree of each stratum of the form σ_{α}. In the rational case g = 0, knowing these degrees and the expressions of these strata in terms of the pushforwards of the basic ones, we are able to compute the degrees of at least some of these pushforwards. In our case, it is easy to compute the degrees deg(1), deg(δ_{0,0}), deg(δ_{1,0}), deg(ξ_{2}). Note, however, that these degrees are not sufficient for the computation of the degrees of the multisingular strata: their decomposition contains the class δ_{0,0}^{2}. The degree of the latter can be deduced form that of the stratum σ_{2^{1},2^{1}}, which we know due to the work of Zvonkine [21].

4.3 On nonisolated singularities

Although we do not yet have a complete picture of what happens when nonisolated singularities are included, some words are in order. Consider the first case where the nonisolated singularities give nontrivial contribution, namely the case of codimension 3 classes for g = 0. Denote by σ_{α_{1},α_{2},...}^{exp} the expected expression for the corresponding class obtained by the universal formulas. It differs from the actual class σ_{α_{1},α_{2},...} by some terms supported on the locus I_{∞} ⊂ B of functions with non-isolated singularities. Since the correction term has the same codimension as the subvariety I_{∞}, it should be proportional to the fundamental class of I_{∞} with some coefficient. It is natural to assume that this coefficient is universal, it depends neither on the irreducible component of I_{∞}, nor on n. The numerical manifestation of this assumption is the ‘unexpected’ observation that the degree of the difference

deg(σ_{α,α_{2},...} − σ_{α,α_{2},...}^{exp})

is proportional to the degree of the stratum I_{∞},

deg(I_{∞}) = \frac{1}{8}(n − 1)(n^{3} + 11n^{2} + 34n − 120)n^{n−5},

with integer coefficient. The degrees of σ_{α,α_{2},...}^{exp} are computed by the methods of this paper, and the actual degree of the class σ_{α,α_{2},...} can be derived from the known Hurwitz numbers. In the following table we present the conjectural contribution of nonisolated singularities for some codimension 3 strata:

σ_{4^{1}} = σ_{4^{1}}^{exp} + 5 I_{∞}
σ_{3^{1}1^{1}} = σ_{3^{1}1^{1}}^{exp} − 16 I_{∞}
σ_{2^{2}} = σ_{2^{2}}^{exp} − 9 I_{∞}
σ_{2^{1}1^{2}} = σ_{2^{1}1^{2}}^{exp} + 36 I_{∞}
σ_{1^{4}} = σ_{1^{4}}^{exp} − 16 I_{∞}.

References

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

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

[3] W. Fulton Intersection theory, Springer, Berlin (1998)

[4] M. Giusti Classification des singularités isolées simple d’intersections complètes, Proceedings of Symposia in Pure Mathematics, vol. 40, Part 1, 457–494 (1983)

[5] V. V. Goryunov Singularities of projections of complete intersections, (Russian) in: Current problems in mathematics, Vol. 22, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 167–206 (1983)

[6] I. P. Goulden, D. M. Jackson Transitive factorisation into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc., 125, no. 1, 51– 60 (1997)

[7] J. Harris, D. Mumford On the Kodaira dimension of the moduli space of curves, Invent. Math., 67, no. 1, 23-88 (1982)

[8] A. Hurwitz Über Riemann’sche Flächen mit gegebenen Verzweigungpunkten, Math. Ann., 39, 1–61 (1891)

[9] A. Hurwitz Über die Ancal der Riemann’sche Flächen mit gegebenen Verzweigungpunkten, Math. Ann., 55, 51–60 (1902)

[10] M. E. Kazaryan Multisingularities, cobordisms, and enumerative geometry, Russ. Math. Surveys, 58, no. 4 665–724 (2003)

[11] M. E. Kazaryan Morin maps and their characteristic classes, preprint (2002)

[12] M. E. Kazaryan Classifying spaces of singularities and Thom polynomials, in: New developments in singularity theory, NATO Sci. Ser. II Math. Phys. Chem., 21, Kluwer Acad. Publ., Dordrecht, 117–134 (2001)

[13] M. E. Kazaryan Relative Morse theory of one-dimensional foliations, and cyclic homology, Funct. Anal Appl., 31 no. 1, 20–31 (1997)

[14] S. K. Lando Ramified coverings of the two-dimensional sphere and intersection theory in spaces of meromorphic functions on algebraic curves, Russ. Math. Surv., 57, no. 3, 463–533 (2002).

[15] S. K. Lando, D. Zvonkine On multiplicites of the Lyashko–Looijenga mapping on strata of the discriminant, Func. Anal. Appl.. 33 178–188 (1999)

[16] S. K. Lando, D. Zvonkine Counting ramified coverings and intersection theory on spaces of rational functions I, math.AG/0303218, 26 pp. (2003)

[17] Yu. Manin, P. Zograf, Invertible cohomological field theories and Weil-Petersson volumes, Ann. Inst. Fourier (Grenoble), 50, no. 2, 519-535 (2000)

[18] D. Mumford Towards an enumerative geometry on the moduli spaces of curves, in: Progress in Math., 36, Birkhüaser, Boston MA, 271–328 (1983)

[19] S. M. Natanzon, V. Turaev A compactification of the Hurwitz space, Topology, 38, 889–914 (1999)

[20] D. Zvonkine Multiplicities of the Lyashko–Looijenga map on its strata, C. R. Acad. Sci, t. 324, série I, p. 1349–1353 (1997)

[21] D. Zvonkine Counting ramified coverings and intersection theory on Hurwitz spaces II (Local structures of Hurwitz spaces and combinatorial results), preprint, math.AG/0304251 39 pp. (2003)