COINCIDENT ROOT LOCI OF BINARY FORMS

L. M. FEHÉR, A. NÉMETHI, AND R. RIMÁNYI

Abstract. Coincident root loci are subvarieties of S^{d}ℂ^{2} — the space of binary forms of degree d — labelled by partitions of d. Given a partition λ, let X_{λ} be the set of forms with root multiplicity corresponding to λ. There is a natural action of GL_{2}(ℂ) on S^{d}ℂ^{2} and the coincident root loci are invariant under this action. We calculate their equivariant Poincaré duals generalizing formulas of Hilbert and Kirwan. In the second part we apply these results to present the cohomology ring of the corresponding moduli spaces (in the GIT sense) with geometrically defined relations.

1. Introduction

Consider the d-th symmetric power S^{d}ℂ^{2} of the standard representation of GL_{2}(ℂ), that is the action of G on the space V_{d} of degree d homogeneous polynomials in two variables x, y. For any partition λ = (λ_{1}, λ_{2}, …, λ_{n}) of d (i.e. ∑_{j} λ_{j} = d) we define

X_{λ} = {B(x, y) ∈ V_{d} | B = ∏_{j=1}^{n} L_{j}^{λ_{j}} for some linear forms L_{j}},

which is a subvariety invariant under the group action. It is called the coincident root loci associated with λ. Clearly, it is a cone in V_{d}, let ℙX_{λ} be its projectivization in the projective space ℙV_{d}. In fact, it is more convenient to use a different notation for partitions: λ = (1^{e_{1}}2^{e_{2}} … r^{e_{r}}) will mean the partition consisting of e_{1} copies of 1, e_{2} copies of 2, etc. Then ∑ ie_{i} = d, ∑ e_{i} = n and the complex dimension of ℙX_{λ} is exactly n.

The study of coincident root loci probably started with Cayley. E.g., the very first question of this type asks the characterization of polynomials B with a double root. The answer is clearly the vanishing of the discriminant which provides in this way an equation for X_{(1^{d−2}2)}. For higher codimensional coincident root loci finding the defining equations is very complicated (see [Chi01] for recent results). However, important geometric information can be obtained about these subvarieties. E.g., the starting point of the present paper was Hilbert’s formula which calculates the degree of ℙX_{λ} ⊂ ℙV_{d}:

deg(ℙX_{λ}) = \frac{n!}{∏_{i}(e_{i}!)} ∏_{i} i^{e_{i}}.

We can interpret this formula as follows: for a generic family of polynomials parametrized by a projective space of dimension equal to the codimension of ℙX_{λ} the number of polynomials in the family with root multiplicity λ is deg(ℙX_{λ}).

Supported by FKFP0055/2001 (1st and 3rd author), OTKA T029759 (3rd author) and NSF grant DMS-0088950 (2nd author)

Keywords: Classes of degeneracy loci, Thom polynomials, global singularity theory AMS Subject classification 14N10, 57R45.

Generalizing this we arrive to the theory of degeneracy loci. Suppose we have a vector bundle E → M with fiber S^{d}ℂ^{2} and a generic section s : M → E. Let s^{−1}(X_{λ}) be the set of points in M where the value of s is in X_{λ}. Its Poincaré dual [s^{−1}(X_{λ})] ∈ H^{∗}(M) measures the “size” of s^{−1}(X_{λ}). It turns out that for any S^{d}ℂ^{2}-bundle, [s^{−1}(X_{λ})] can be deduced from the corresponding cohomology class of the universal bundle associated with the GL_{2}(ℂ)– representation S^{d}ℂ^{2}. This universal invariant is called the GL_{2}(ℂ)-equivariant Poincaré dual, or Thom polynomial of X_{λ} in S^{d}ℂ^{2}. In section 3 we determine all these polynomials.

Calculating equivariant Poincaré duals for invariant subvarieties of representations has a long history. We can interpret many results of the nineteenth century algebraic geometers in these terms. From the 1970’s the main method was a type of resolution of the subvariety, initiated by Porteous [Por71]. The method requires a deep understanding of the geometry of the resolution and can be carried out only in special cases. Most examples can be found in [Ful98]. The first and third author designed a different method (the method of restriction equations, see [FRa]) based on ideas coming from calculating Thom polynomials in singularity theory [Rim01]. However the method of restriction equations works well mainly if the representation has finitely many orbits which is usually not the case (e.g. for S^{d}ℂ^{2} if d > 3).

In this paper we return to the technique of resolution, however in a very different way. The main novelty is that our new approach requires only knowledge of some basic cohomological data. Consequently, the method is more flexible. We illustrate this method here by the coincident root loci, but the range of applications is much wider. (For example, in a forthcoming paper we plan to discuss the case of loci of reducible hypersurfaces.)

Parallel to our work B. Kőműves also provided a presentation of these Poincaré duals in a completely different form [Kőm03]. He worked more in the spirit of the method of restriction equations studying incidences of the coincident root loci with the orbits X_{(i,d−i)}. We are in the process of comparing these formulas.

In section 4 we study the cohomology ring of the moduli space of the representation S^{d}ℂ^{2} (in the Geometric Invariant Theory sense). Following the paper of Atiyah and Bott [AB83] a whole theory for calculating cohomology rings of the moduli space of representations was built up by F. Kirwan; as well as more algebraic methods were successfully applied by e.g. M. Brion, S. Martin. However, the application of the general theorems to specific examples is often not easy. Our approach results explicit presentations of the rational cohomology rings H_{G}^{∗}(X^{ss}), H^{∗}(X^{ss}/G) and H_{G}^{∗}(X^{s}) ≅ H^{∗}(X^{s}/G) in terms of generators and relations (if d is odd then all these rings coincide, but for the even case they are different). We wish to emphasize that a main advantage of our presentation of the cohomology rings is that we attribute to the set of relations deep geometric significance: they are the universal Thom polynomials (Poincaré duals) of some distinguished spaces X_{λ}.

2. Review on affine and projective Thom polynomials

Let the group G act on the complex vector space V, and let η be an invariant variety in V, which supports a fundamental class (for more details see [FRa]). Then define the (affine) Thom polynomial of η as the Poincaré dual of the fundamental homology class of η in equivariant cohomology:

Tp_{η} = Poincaré dual of [η] ∈ H_{G}^{∗}(V, ℤ).

The vector space V is contractible, hence the ring H_{G}^{∗}(V, ℤ) is naturally isomorphic to H^{∗}(BG, ℤ), the ring of G-characteristic classes. The degree of Tp_{η} is the real codimension 2c of η in V, hence Tp_{η} ∈ H^{2c}(BG, ℤ). The direct geometric meaning of Tp_{η} is the following.

Consider a fiber bundle ξ with fiber V and structure group G over a manifold M. Because of its invariance, the set η can be defined in each fiber, let the union of these be η(ξ). Then consider those points where a generic section s of ξ hits η, that is s^{−1}(η(ξ)) ⊂ M. By Poincaré duality this set defines a cohomology class in M. Standard arguments show that this class equals Tp_{η}(ξ) := f_{ξ}^{∗} Tp_{η}, where f_{ξ} : M → BG is a classifying map of ξ.

We will also use the projective version of Thom polynomials (see [FNR]), as follows. Assume that G acts on V in such a way that the scalars are in the image of G → GL(V). Then the orbits of this action (different from {0}) are in bijection with the orbits of the induced action of G an ℙV. Also, the corresponding orbits, η and ℙη have the same codimension. The equivariant Poincaré dual of ℙη will be called the projective Thom polynomial of η:

ℙ Tp_{η} = Poincaré dual of [ℙη] ∈ H_{G}^{∗}(ℙV, ℤ) = H^{∗}(BG, ℤ)[x]/(Q(x)) (deg(x) = 2),

where Q(x) is the product of all (x + α_{j})’s, where α_{j} ∈ H^{2}(BG) are the weights of the representation of G on V [BT82]. The projective Thom polynomial can be written as ℙ Tp_{η} = p_{c} +p_{c−1}x+…+p_{0}x^{c}, where p_{i} ∈ H^{2i}(BG). By [FNR, Section 6], p_{c} = Tp_{η} and p_{0} is the degree of the variety ℙη. Seemingly, the projective Thom polynomial contains more information then the “affine” one. This is not the case: ℙ Tp_{η} can be obtained from Tp_{η} by a simple substitution, see Theorem 6.1 in [FNR] (although this fact will not be used in the present paper). In particular, the degree p_{0} of ℙη itself can be obtained from Tp_{η} by a substitution. For this substitution in our specific case, see 3.9(2).

3. Coincident root loci

Consider the d-th symmetric power V_{d} = S^{d}ℂ^{2} of the standard representation of G = GL_{2}(ℂ), and the invariant subvariety X_{λ} associated with a partition λ = (λ_{1}, λ_{2}, …, λ_{n}) of d (cf. introduction). In this section we compute its Thom polynomial Tp_{λ} ∈ H^{∗}(BG, ℤ).

Points in the projectivization ℙV_{d} of V_{d} can be identified with d-tuples of points in ℙ^{1} = {(x : y)} (counted with multiplicities). The projectivization ℙX_{λ} is then the closure of the set of d-tuples having n distinct points with multiplicities λ_{1}, λ_{2}, …, λ_{n}. The variety ℙX_{λ} is called the coincident root loci.

Consider also the other notation λ = (1^{e_{1}}2^{e_{2}} … r^{e_{r}}) with ∑ ie_{i} = d and ∑ e_{i} = n (cf. introduction). Then ℙX_{λ} is the image of the map

ϕ : ℙV_{e_{1}} × ℙV_{e_{2}} × … × ℙV_{e_{r}} → ℙV_{d}

defined (via point-tuples of ℙ^{1}) by (D_{1}, D_{2}, …, D_{r}) ↦ ∑ iD_{i}. It is readily seen that ϕ is birational onto its image ℙX_{λ} (i.e. is a resolution of ℙX_{λ}). In particular, dim(ℙX_{λ}) = n and Tp_{λ} is of degree d − n (cf. section 2).

The map ϕ is equivariant under the action of G on the two spaces, hence it makes sense to talk about the maps ϕ^{∗} (induced by ϕ) and ϕ_{!} (the push-forward map of ϕ) in G-equivariant cohomology. The equivariant cohomology rings are as follows (cf. e.g. [BT82], p. 270):

H_{G}^{∗}(_{∏i} ℙV_{e_{i}}, ℤ) = R[x_{1}, …, x_{r}]/(Q_{e_{1}}(x_{1}), …, Q_{e_{r}}(x_{r})), resp. H_{G}^{∗}(ℙV_{d}, ℤ) = R[x]/(Q_{d}(x)).

Here

R = H^{∗}(BG, ℤ) = ℤ[c_{1}, c_{2}] = ℤ[u, v]^{ℤ_{2}},

where ℤ_{2} permutes the roots u and v (hence c_{1} = u + v and c_{2} = uv); and the polynomial Q_{k} (k ≥ 1) is defined by

Q_{k}(y) = ∏_{α is a weight of S^{k}ℂ^{2}}(y + α) = ∏_{j=0}^{k}(y + ju + (k − j)v).

The map ϕ^{∗} is a ring homomorphism, it leaves elements of R invariant, and it maps x to

ϕ^{∗}(x) = ∑_{i=1}^{r} ix_{i}.

The above rings can be described also as finite dimensional modules over R, spanned by ∏_{i} x_{i}^{k_{i}}(0 ≤ k_{i} ≤ e_{i}) and x^{k}(0 ≤ k ≤ d), respectively. A representative of an element [f] (in any of these rings) is reduced if it is written as an R-linear combination of these monomials. It is denoted by [f]_{red}. In this language, the value of the integration maps (along the fibers)

∫_{∏ ℙV_{e_{i}}} : H_{G}^{∗}( ∏_{i} ℙV_{e_{i}}, ℤ) → R, resp. ∫_{ℙV_{d}} : H_{G}^{∗}(ℙV_{d}, ℤ) → R

are the coefficients of the top degree monomials in the corresponding reduced forms: i.e. the coefficient of x^{e} := ∏_{i} x_{i}^{e_{i}} in the first case, and the coefficient of x^{d} in the second case.

Set q(x) := (Q_{d}(x) − C_{d+1})/x = x^{d} + C_{1}x^{d−1} + … + C_{d}, where Q_{d}(x) = ∑_{j=0}^{d+1} C_{d+1−j}(c_{1}, c_{2})x^{j} and C_{0} = 1. Theorem 3.1. Tp_{λ} equals ∫_{∏ ℙV_{e_{i}}} ϕ^{∗}(q).

Proof. First we prove that Tp_{λ} = ∫_{ℙV_{d}}(q · ℙTp_{λ}). Indeed, from the general theory of projective and affine Thom polynomials (cf. section 2) we know that ℙTp_{λ} = p_{d−n}+p_{d−n−1}x+…+p_{0}x^{d−n}, where p_{j} ∈ ℤ[c_{1}, c_{2}] and p_{d−n} = Tp_{λ}. When we multiply x^{j}p_{d−n−j} (1 ≤ j ≤ d − n) with q = (Q_{d} − C_{d+1})/x and reduce it modulo Q_{d}(x), the coefficient of x^{d} will be 0. So the only contribution comes from qp_{d−n}, which is the coefficient p_{d−n} of p_{d−n}x^{d}.

Now, using the definition of ℙTp_{λ} and the fact that ϕ is birational, we have Tp_{λ} = ∫_{ℙV_{d}}(q · ϕ_{!}(1)) which equals ∫_{∏ ℙV_{e_{i}}} ϕ^{∗}(q), what we wanted to prove.

Theorem 3.1 gives the following computational recipe: T p_{λ} is the top coefficient (i.e. the coefficient of x^{e}) of ϕ^{∗}(q)_{red}. Notice that any representative [f]_{red} is automatically computed by computer algebra packages (e.g. [GS]), hence one gets an algorithmic solution of finding the Thom polynomials, see e.g. www.math.ohio-state.edu/˜ rimanyi/progs/rootloci.m2. We can, however, give very explicit formulae as well.

Formulae for Thom polynomials.

Lemma 3.2. Set f ∈ R[y] with class [f] mod Q_{e}(y). Then the top coefficient of the reduced representative [f]_{red} is

(1)  ∫_{ℙV_{e}} [f] = \frac{1}{(v − u)^{e}} ∑_{s=0}^{e} \frac{(−1)^{s}f( − (e − s)u − sv)}{s!(e − s)!}.

Proof. Since the formula is linear in f, it is enough to verify it for any f(y) = y^{j} (j ≥ 0). In this case we need A_{e} where y^{j} ≡ A_{e}y^{e} + A_{e−1}y^{e−1} + … + A_{0} modulo the ideal (Q_{e}(y)). If we consider this congruence for y = −eu, −(e − 1)u − v, …, −ev then we get a system of equations for A_{e}, …, A_{0} (since Q_{e}(y) vanishes at these points). The matrix of this system is a Vandermonde matrix, so by Cramer’s rule we get the formula.

Corollary 3.3. (The “naive” formula) Let ∑_{s_{1},...,s_{r}} denote the sum over 0 ≤ s_{i} ≤ e_{i} for each 1 ≤ i ≤ r. Then

Tp_{λ} = \frac{1}{(v − u)^{n}} ∑_{j=n}^{d} ∑_{j_{1}+...+j_{r}=j} C_{d−j}\binom{j}{j_{1}, …, j_{r}} ∑_{s_{1},...,s_{r}} ∏_{i=1}^{r} \frac{(−1)^{s_{i}}(−i)^{j_{i}}((e_{i} − s_{i})u + s_{i}v)^{j_{i}}}{s_{i}!(e_{i} − s_{i})!} .

Proof. Write ∑_{j=n}^{d} C_{d−j}(∑ ix_{i})^{j} as a linear combination of monomials of type ∏_{i=1}^{r} x_{i}^{j_{i}}. The polynomial Q_{e_{i}}(x_{i}) only contains the variable x_{i}. Hence to find the top coefficient of the remainder of ∏_{i} x_{i}^{j_{i}}, we can simply multiply the top coefficient of the remainders of x_{i}^{j_{i}} modulo Q_{e_{i}}(x_{i}). Therefore, the formula follows from lemma 3.2 applied one-by-one for each x_{i}^{j_{i}}.

One can get a more interesting formula as follows. First notice that xq + C_{d+1} = Q_{d}, hence (∑ ix_{i})ϕ^{∗}(q) ≡ −C_{d+1} modulo the ideal I ⊂ R[x_{1}, …, x_{r}] generated by all Q_{e_{i}}(x_{i}) (1 ≤ i ≤ r). We consider the following identities regarding 1/ ∑_{i} ix_{i}. Let t be a free variable. Then

\frac{1}{−t + ∑_{i} ix_{i}} = \frac{1}{−t} ∑_{j≥0} (_{∑i}ix_{i}/t)^{j} = \frac{1}{−t} ∑_{j≥0} ∑_{j_{1}+...+j_{r}=j} \binom{j}{j_{1}, …, j_{r}}_{∏i}(ix_{i}/t)^{j_{i}}.

By lemma 3.2, the top coefficient of the last expression is

\frac{1}{−t} ∑_{j≥0} ∑_{j_{1}+...+j_{r}=j} \binom{j}{j_{1}, …, j_{r}} ∏_{i} ∑_{s_{i}=0}^{e_{i}} \frac{(−1)^{s_{i}}}{(v − u)^{e_{i}}s_{i}!(e_{i} − s_{i})!}( (e_{i} − s_{i})u + s_{i}v )^{j_{i}}(−i/t)^{j_{i}}

= \frac{1}{(−t)(v − u)^{n}} ∑_{s_{1},...,s_{r}} \frac{(−1)^{∑_{i} s_{i}}}{∏_{i} s_{i}!(e_{i} − s_{i})!} · \frac{1}{1 + ∑_{i} i( (e_{i} − s_{i})u + s_{i}v)/t}

= \frac{1}{(v − u)^{n}} ∑_{s_{1},...,s_{r}} \frac{(−1)^{∑_{i} s_{i}}}{∏_{i} s_{i}!(e_{i} − s_{i})!} · \frac{1}{−t − du + (∑_{i} is_{i})(u − v)}.

Let A(t) be this last expression. The above identities show the following congruence (valid for generic t):

(1)  (−t +_{∑i}ix_{i})( A(t)x^{e} + lower order terms ) ≡ 1 (mod I).

Evidently, this is true for t = 0 as well. On the other hand, notice that there is a unique reduced Y ∈ R[x_{1}, …, x_{r}] satisfying (∑ ix_{i})Y ≡ −C_{d+1} (mod I). Indeed, if both Y and Y′ satisfy it, then −C_{d+1}Y′ ≡ Y (∑ ix_{i})Y′ ≡ −C_{d+1}Y, hence Y = Y′. Since (∑ ix_{i})ϕ^{∗}(q) ≡ −C_{d+1}, from (1) (with t = 0) we get that the top coefficient of ϕ^{∗}(q)_{red} is −C_{d+1}A(0). Hence, we proved:

Theorem 3.4. With the notation C_{d+1} := C_{d+1}(S^{d}ℂ^{2}) = ∏_{j=0}^{d}( ju + (d − j)v), one has

Tp_{λ} = \frac{C_{d+1}}{(v − u)^{n}} · ∑_{s_{1},...,s_{r}} \frac{(−1)^{∑_{i} s_{i}}}{∏_{i} s_{i}!(e_{i} − s_{i})!} · \frac{1}{du − (∑_{i} is_{i})(u − v)}.

This can also be considered as a higher order divided difference formula, cf. 3.7.

Example 3.5. If λ = i^{e_{i}}, hence d = ie_{i}, then

Tp_{λ} = i^{e_{i}} · ∏_{0≤j≤d;i∤j} ( ju + (d − j)v).

This can be deduced from 3.4 (cf. with the next remark), but one also can argue as follows. Since ix_{i}ϕ^{∗}(q)+C_{d+1} ≡ 0 (mod Q_{e_{i}}(x_{i})), clearly ix_{i}ϕ^{∗}(q)_{red}+C_{d+1} ≡ 0 as well. Since ix_{i}ϕ^{∗}(q)_{red}+C_{d+1} and Q_{e_{i}}(x_{i}) both have degree e_{i} + 1, one gets that ix_{i}ϕ^{∗}(q)_{red} + C_{d+1} = C · Q_{e_{i}}(x_{i}) for some C ∈ R. Comparing the coefficients of x_{i}^{e_{i}+1} and x_{i}^{0}, one obtains

i Tp_{λ} = C_{d+1}(S^{d}ℂ^{2})/C_{e_{i}+1}(S^{e_{i}}ℂ^{2}).

Remark 3.6. Lemma 3.2 has the following consequence. For some C ∈ R and g ∈ R[y], we denote by [C/g]_{red} (or by ∫_{ℙV_{e}}[C/g]) that reduced element which satisfies [C/g]_{red} · g ≡ C (mod Q_{e}(y)) (if it exists). Then one also has:

(1)  ∫_{ℙV_{e}} [C/g] = \frac{1}{(v − u)^{e}} ∑_{s=0}^{e} \frac{(−1)^{s}}{s!(e − s)!} · \frac{C}{g( − (e − s)u − sv)}.

Its proof is similar to the proof of 3.4, which, in fact, is a multivariable version of (1) (applied for −C_{d+1}/ ∑ ix_{i}).

Let us consider again λ = i^{e_{i}}. Theorem 3.4 and (1) gives that Tp_{λ} = ∫_{ℙV_{e_{i}}}[−C_{d+1}(S^{d})/ix_{i}]. But x_{i}(x_{i}^{e_{i}} + ⋯) + C_{e_{i}+1}(S^{e_{i}}) = Q_{e_{i}}, hence ∫_{ℙV_{e_{i}}}[−C_{e_{i}+1}(S^{e_{i}})/x_{i}] = 1. In particular, Tp_{λ} = C_{d+1}(S^{d})/iC_{e_{i}+1}(S^{e_{i}}), as it was verified in 3.5.

Example 3.7. Assume that λ = i^{e_{i}}j^{e_{j}} (i ≠ j). Consider the expression given by 3.4 for this λ, and apply in variable x_{i} the identity 3.6(1). Clearly du−(is_{i} +js_{j})(u−v) = g(−e_{i}u+s_{i}(u−v)), where g(x_{i}) := a − ix_{i} with a := je_{j}u − js_{j}(u − v). Therefore

Tp_{λ} = \frac{C_{d+1}(S^{d})}{(v − u)^{e_{j}}} ∑_{s_{j}=0}^{e_{j}} \frac{(−1)^{s_{j}}}{s_{j}!(e_{j} − s_{j})!} · ∫_{ℙV_{e_{i}}} [1/g(x_{i})].

Since Q_{e_{i}}(x_{i}) − Q_{e_{i}}(a/i) = (x_{i} − a/i)(x_{i}^{e_{i}} + ⋯) one gets ∫_{ℙV_{e_{i}}}[iQ_{e_{i}}(a/i)/g(x_{i})] = 1. Hence

Tp_{λ} = \frac{C_{d+1}(S^{d})}{(v − u)^{e_{j}}} ∑_{s_{j}=0}^{e_{j}} \frac{(−1)^{s_{j}}}{s_{j}!(e_{j} − s_{j})!} · \frac{1}{i · Q_{e_{i}}( (je_{j}u − js_{j}(u − v) )/i)}.

For example, assume that λ = i^{e_{i}}j, i.e. e_{j} = 1. Then s_{j} = 0 or 1, hence

Tp_{λ} = \frac{C_{d+1}(S^{d})}{i(v − u)} · (\frac{1}{Q_{e_{i}}(ju/i)} − \frac{1}{Q_{e_{i}}(jv/i)}).

It is convenient to express this in the language of divided difference: If P (u, v) is a polynomial in two variables (u, v), we denote by ∂(P) the polynomial (P (u, v) − P (v, u))/(u − v). Then

Tp_{(i^{e_{i}}j)} = \frac{1}{i} · ∂ ( \frac{C_{d+1}(S^{d})}{Q_{e_{i}}(jv/i)} ) = i^{e_{i}} · ∂ ( ∏ ( (d − k)v + ku )),

where the product is over k with 0 ≤ k ≤ d, but k ≠ is with 0 ≤ s ≤ e_{i}. In particular,

(1)  Tp_{(1^{e_{1}}j)} = ∂ ( ∏_{l=0}^{j−1} ( lv + (e_{1} + j − l)u )) (for j ≥ 2),

which is equivalent with Kirwan’s formula [Kir92, page 902].

Example 3.8. Assume that d = 2h is even, h > 2 and λ = (1^{h−j}, j, h) for some 1 < j < h. By a similar argument as in 3.7 and by a computation, one has

Tp_{λ} = \frac{C_{d+1}(S^{d})}{(u − v)^{2}} · (\frac{1}{Q_{h−j}(hu + ju)} − \frac{1}{Q_{h−j}(hu + jv)} − \frac{1}{Q_{h−j}(hv + ju)} + \frac{1}{Q_{h−j}(hv + jv)})

= ∂[ \frac{C_{d+1}(S^{d})}{u − v} · (\frac{1}{Q_{h−j}(hv + jv)} − \frac{1}{Q_{h−j}(hv + ju)})] = ∂ [D_{j} · ∏_{l=0}^{h−1} ( lv + (d − l)u )],

where

D_{j} := \frac{1}{u − v} · [ ∏_{l=h−j+1}^{h}( lv + (d − l)u) − ∏_{l=0}^{j−1}( lv + (d − l)u)].

E.g., if j = 2, then

Tp_{λ} = h(h − 1) · ∂ [(u + 3v) ∏_{l=0}^{h−1} ( lv + (d − l)u )].

Remarks 3.9. (1) The Thom polynomials are connected by many interesting polynomial relations. E.g., the next section presents two situations when the ideal generated by natural families of Thom polynomials is generated only by two of them. Some of these relations can be verified easily. E.g., assume d = 2h as in 3.8, consider the partitions λ′_{0} = (1^{h−2}, 2, h), λ_{0} = (1^{h}, h), λ_{1} = (1^{h−1}, h + 1) and λ_{2} = (1^{h−2}, h + 2). Then from 3.7(1) and 3.8, one gets Tp_{λ_{1}} = hc_{1} · Tp_{λ_{0}} and

(h − 1) · Tp_{λ_{2}} = (h − 1)(h − 2)c_{1} · Tp_{λ_{1}} +c_{1} Tp_{λ′_{0}} .

(2) Using [FNR], one can determine deg(ℙX_{λ}) by the substitution u = v = 1/d in Tp_{λ} ∈ ℤ[u, v]. The interested reader is invited to verify the compatibility of Hilbert’s result (cf. introduction) with this section.

(3) In the sequel we will use many times the following divided difference formula. For any polynomial A ∈ ℚ[u, v] write A^{∗}(u, v) := A(v, u). Then

∂(AB) = B^{∗} · ∂(A) + A · ∂(B).

4. Thom polynomial description of the cohomology ring of the moduli space

In this section we apply the coincident root loci formulas in the study of the cohomology ring of the moduli space of the representation S^{d}ℂ^{2} (in the Geometric Invariant Theory sense). We calculate the rational cohomology rings H_{G}^{∗}(X^{ss}), H^{∗}(X^{ss}/G) and H_{G}^{∗}(X^{s}) ≅ H^{∗}(X^{s}/G) in terms of generators and relations. If d is odd then all these rings coincide, but for the even case they are different.

There is an extensive literature on these cohomology rings, both from combinatorial-algebraic (see e.g. [Bri91], [Mar99]) and from geometric point of view (the Atiyah-Bott-Kirwan theory [Kir84]). Our approach (in the odd d case) is closest to that of Kirwan. The advantage of our approach is that we treat the odd and even cases in a uniform language, and that we provide for the above cohomology rings a very transparent structure: we obtain explicit presentations of them in terms of generators and relations with clear geometric meanings.

Let us sketch our approach in the odd case first (for details see below). In this case the Kirwan stratification of S^{d}ℂ^{2} is G-perfect since the normal (equivariant) Euler classes of the strata are not zero-divisors. It implies that the spectral sequence of the corresponding filtration degenerates. It is not difficult to calculate all but the 0^{th} column of the E_{1}-table, so by subtraction we can calculate the ranks of the 0^{th} column: the Betti numbers of H_{G}^{∗}(X^{ss}). Also by G-perfectness the natural map

κ : H_{G}^{∗}(S^{d}ℂ^{2}) ≅ ℚ[c_{1}, c_{2}] → H_{G}^{∗}(X^{ss})

is surjective so we need to find relations in terms of c_{1} and c_{2}, ie. we have to find generators of Ker(κ). If Y ∩ X^{ss} = ∅ for an invariant subvariety Y then clearly [Y] ∈ Ker(κ). (This idea was studied in [FRb]). So all the higher Kirwan strata provide relations. But the Kirwan strata are coincident root loci for specific partitions and we can calculate their equivariant Poincaré dual using the first part of the paper. It turns out that the first two Kirwan strata are enough to generate Ker(κ) which can be checked by a simple Betti number calculation.

The main difficulty in the even case is that for one of the strata in a refined Kirwan stratification the normal (equivariant) Euler class is a zero-divisor. To prove G-perfectness we use the results of the first part of the paper. Namely we show that certain elements in the E_{1}-table can be represented by the Poincaré dual of coincident root loci (these are not Kirwan strata!) and they survive to E_{∞}, hence they could not be hit by a differential. After G-perfectness is proven the process is the same as in the odd case. We can find coincident root loci in the null cone such that their Poincaré dual generate Ker(κ). Here we also need two coincident root loci but one of them is not a Kirwan stratum.

In this section all cohomologies are meant with rational coefficients.

Let us consider the Kirwan-stratification (see [Kir92] and [Kir84]) of the vector space V_{d}: • X^{ss} = {B | B has no root of multiplicity > d/2},

• X_{i} = {B | B has a root of multiplicity i but no with multiplicity i + 1} (d/2 < i ≤ d),

• X_{0} = {0}. The strata are smooth open submanifolds, the complex codimensions are 0, i − 1, d + 1 in the three cases. By F_{i} = ∪ strata of complex codimension ≤ i we get a filtration of V_{d}:

∅ = F_{−1} ⊂ F_{0} ⊂ F_{1} ⊂ … ⊂ F_{d+1} = S^{d}ℂ^{2}.

Let E_{∗}^{∗,∗} be the associated spectral sequence in G-equivariant cohomology with ℚ-coefficients.

Proposition 4.1.

(1) E_{1}^{0,∗} = H_{G}^{∗}(X^{ss}; ℚ)

(2) E_{1}^{2p,∗} = H^{∗}(BU(1); ℚ) for p = [d/2], …, d − 1;

(3) E_{1}^{2(d+1),∗} = H^{∗}(BG; ℚ);

(4) E_{1}^{∗,∗} = 0 for all cases not covered by (1), (2), (3);

(5) The spectral sequence converges to H^{∗}(BG; ℚ);

(6) The spectral sequence degenerates at E_{1}^{∗,∗} (in particular, H_{G}^{odd}(X^{ss}, ℚ) = 0).

Proof. By definition we have E_{1}^{2p,∗} = H_{G}^{2p+∗}(F_{p}, F_{p−1}) which is by Thom isomorphism H_{G}^{∗}(F_{p} \ F_{p−1}). This proves (1) and (4). For p = d + 1 we have E_{1}^{2(d+1),∗} = H_{G}^{∗}({0}) = H^{∗}(BG) which proves (3). For d/2 < i ≤ d we define Y_{i} = {B ∈ X_{i} : x^{i}|B and coeff (x^{i}y^{d−i}) = 1}. Let H be

the stabilizer subgroup of Y_{i}, i.e. the group of matrices of the form (\begin{matrix}α_{1} & β \\ 0 & α_{2} \end{matrix}) with α_{1}^{i}α_{2}^{d−i} = 1.

Since Y_{i} is contractible, and X_{i} = G ×_{H} Y_{i}, part (2) follows from

H_{G}^{∗}(X_{i}) ≅ H_{G}^{∗}(G ×_{H} Y_{i}) ≅ H_{H}^{∗}(Y_{i}) ≅ H^{∗}(BH) ≅ H^{∗}(BU(1)) (over ℚ).

The degeneracy of the spectral sequence—called G-perfectness by Atiyah-Bott in [AB83]— follows from usual arguments, as follows. Let us build up V_{d} by gluing the strata one by one together in order of increasing codimension. Then at one step we have U and glue a new stratum X of complex codimension c to it. We need to prove that the first map in the diagram

H_{G}^{n−2c}(X) ≅ H_{G}^{n}(U ∪ X, U) → H_{G}^{n}(U ∪ X) → H_{G}^{n}(X)

is injective. However, the whole composition is the multiplication with the equivariant Euler class of the stratum X. This is an injective map being a multiplication by a non-zero element in a polynomial ring. (For a computation of an equivariant Euler class see the proof of 4.7.)

Since E_{∞} = E_{1}, the sum of the ranks of the groups in diagonal (i.e. p + q = r) entries must be the rank of the appropriate cohomology group of H^{∗}(BG; ℚ). Thus we have the following

Corollary 4.2. Let h := [d/2]. The Poincaré series of the ring H_{G}^{∗}(X^{ss}; ℚ) is

\frac{1}{(1 − t)(1 − t^{2})}(1 − t^{d+1}) − \frac{1}{1 − t}(t^{h} + … + t^{d−1}) = \frac{1 − t^{h} − t^{h+1} + t^{d}}{(1 − t)(1 − t^{2})} (deg(t) = 2).

What we obtained so far is basically equivalent to the Atiyah-Bott-Kirwan theory applied to our representation, see [Kir84, 16.2].

What can also be seen from the spectral sequence is that H_{G}^{∗}(X^{ss}) = H^{∗}(BG)/I where the ideal comes from the p > 0 columns of the spectral sequence. Thus among the elements of I we have the ones that are the images of the generators of E_{1}^{2p,0} under the edge-homomorphism. For [d/2] ≤ p ≤ d − 1, these are exactly the Thom polynomials corresponding to the strata X_{i}, i = p + 1. We have Tp(X_{i}) = Tp_{λ} whit λ = (1^{d−i}, i), since the closures of X_{i} and X_{λ} are the same. The above Betti number computation can be used to test if a few of these Thom polynomials are enough to generate I.

Theorem 4.3. Set λ_{1} = (1^{d−h−1}, h + 1) and λ_{2} = (1^{d−h−2}, h + 2), where h = [d/2]. Then I is generated by Tp_{λ_{1}} and Tp_{λ_{2}}. In particular,

H_{G}^{∗}(X^{ss}; ℚ) = ℚ[c_{1}, c_{2}]/(Tp_{λ_{1}}, Tp_{λ_{2}}).

Proof. We already observed that the given two Tp’s are in I. Now we prove that the ring on the right hand side has the same Poincaré series as the one given in Corollary 4.2.

We claim that the ideal J := (Tp_{λ_{1}}, Tp_{λ_{2}}) has the following R–resolution: 0 ← J ← R(h) ⊕ R(h + 1) ← R(d) ← 0. If d = 2h + 1 then for this we only need to prove that Tp_{λ_{1}} and Tp_{λ_{2}} have no nontrivial common divisor D. We know that Tp_{λ_{1}} = ∂(Π), Tp_{λ_{2}} = ∂(ΠL), where Π(u, v) = ∏_{l=0}^{h}(lv +(d−l)u) and L(u, v) = (h+1)v +hu. By 3.9(3), if D|gcd(Tp_{λ_{1}}, Tp_{λ_{2}}), then D|Π, hence D|gcd(Π, ∂(Π)) as well. But gcd(Π, Π^{∗}) = 1, which ends the proof of the claim.

So we get the Poincaré series of R/J as (1 − t^{h} − t^{h+1} + t^{2h+1})/(1 − t)(1 − t^{2}), which is the same as the Poincaré series of H_{G}^{∗}(X^{ss}; ℚ). For d even the proof is similar.

Discussion 4.4. The cohomology ring of X^{ss}/G. Observe that if d is odd then X^{ss} = X^{s}, and all stabilizers of polynomials in X^{ss} are finite. Therefore, we have the ring isomorphism H_{G}^{∗}(X^{ss}; ℚ) = H^{∗}(X^{ss}/G; ℚ) with Poincaré polynomial (1 − t^{h})(1 − t^{h+1})/(1 − t)(1 − t^{2}).

If d = 2h is even, then X^{ss}/G = X^{s}/G ∪ {p^{ss}}, where p^{ss} is the unique “semisimple point” of X^{ss}/G. The Poincaré series of H_{G}^{∗}(X^{ss}) is infinite; it is:

(1) \frac{1}{1 − t^{2}} + t · P (t), where P (t) is the polynomial \frac{(1 − t^{h−1})(1 − t^{h})}{(1 − t)(1 − t^{2})} (deg(t) = 2).

All the stabilizers of the stable part are finite, and there is only one orbit in the strict semistable part with infinite stabilizer H^{ss}, namely the orbit of the partition (h, h). H^{ss} can be described explicitly, and one has an exact sequence 1 → U(1) × ℤ_{h} → H^{ss} → ℤ_{2} → 1. Hence BH^{ss} is a double covering of BU(1)×Bℤ_{h} with rational cohomology H^{∗}(BH^{ss}) = H^{∗}(BU(1))^{ℤ_{2}} = ℚ[t]^{ℤ_{2}} (deg_{t} = 2). Here the ℤ_{2}-action is t ↦ ±t, hence the invariant part is ℚ[t^{2}] with an infinite Poincaré series 1/(1 − t^{2}). This is exactly the “infinite contribution” in the above Poincaré series of H_{G}^{∗}(X^{ss}).

In fact, the map r : H^{∗}(BG) → H^{∗}(BH^{ss}) (induced by the inclusion) is the following. At the level of roots, it is given by u ↦ ±t and v ↦ ∓t, hence it is the epimorphism r : ℚ[c_{1}, c_{2}] → ℚ[t^{2}] given by c_{1} ↦ 0 and c_{2} ↦ −t^{2}.

As usual, for any connected space Z, let H̃^{∗}(Z) be the kernel of H^{∗}(Z) → H^{∗}(point), as an ideal (or subring without unit) in H^{∗}(Z). The ring H^{∗}(Z) can be reconstructed from H̃^{∗}(Z) by adding the unit: H^{∗}(Z) = ℚ⟨1⟩ ⊕ H̃^{∗}(Z) (with the natural multiplication).

Let o be the orbit corresponding to the partition (h, h) and consider the natural inclusion j : o ×_{G} EG → X^{ss} ×_{G} EG. Obviously, o ×_{G} EG can be identified with BH^{ss}. Moreover, j^{∗} : H_{G}^{∗}(X^{ss}) → H^{∗}(BH^{ss}) induced by j can be identified with the epimorphism ℚ[c_{1}, c_{2}]/(Tp_{λ_{1}}, Tp_{λ_{2}}) → ℚ[t^{2}], c_{1} ↦ 0 and c_{2} ↦ −t^{2} induced by r above. In fact, Tp_{λ_{1}} and Tp_{λ_{2}} are both divisible by c_{1} (cf. 3.9(1)), hence r sends the ideal generated by them to zero.

Finally, notice that H^{∗}(X^{ss} ×_{G} EG, BH^{ss}) = H̃^{∗}(X^{ss} ×_{G} EG/BH^{ss}), and the natural map r : X^{ss} ×_{G} EG/BH^{ss} → X^{ss}/G induces an isomorphism at the level of rational cohomology rings. In particular, the long exact cohomology sequence of the pair (X^{ss} ×_{G} EG, BH^{ss}) transforms into the short exact sequences:

(2)  0 → H̃^{∗}(X^{ss}/G) → H_{G}^{∗}(X^{ss}) →^{j^{∗}} H^{∗}(BH^{ss}) → 0.

Analyzing the kernel of j^{∗}, we get:

Corollary 4.5. With the notations of 4.3, one has the following ring isomorphisms:

H^{∗}(X^{ss}/G; ℚ) = ℚ[c_{1}, c_{2}]/(Tp_{λ_{1}}, Tp_{λ_{2}}) if d is odd;
H^{∗}(X^{ss}/G; ℚ) = ℚ⟨1⟩ ⊕ (c_{1}ℚ[c_{1}, c_{2}])/(Tp_{λ_{1}}, Tp_{λ_{2}}) if d is even.

Notice that the Poincaré series formula 4.4(1) is compatible with the 4.4(2) and 4.5. In particular, if d = 2h, the Poincaré polynomial of H^{∗}(X^{ss}/G) is 1 + tP (t).

Discussion 4.6. The cohomology ring of X^{s}/G. Next, for the case d = 2h, we wish to determine the cohomology ring of the geometric quotient X^{s}/G. In the notations below it is convenient to assume h > 2 (if h = 2, then X^{ss}/G = ℙ^{1}, and X^{s}/G = ℂ).

We consider a similar spectral sequence, but now associated with the stratification

• X^{s} = {B | B has no root of multipicity ≥ h},

• X_{i} = {B | B has exactly one root of multiplicity i but no roots of multiplicity i + 1} (h ≤ i ≤ d),

• o = {the orbit associated with the partition (h, h)},

• X_{0} = {0}.

In lemma 4.1, E_{1}^{0,∗} will be replaced by H_{G}^{∗}(X^{s}). For i > h, the stratum X_{i} is the same as in the previous case. But there are two new strata, namely X_{h} and o. Since o is an orbit with stabilizer H^{ss}, H_{G}^{∗}(o) = H^{∗}(BH^{ss}). The complex codimension of o in V_{d} is d − 2, hence this will provide an additional direct sum contribution in E_{1}^{2(d−2),∗}. Hence, E_{1}^{2p,∗} = H^{∗}(BU(1)) if h ≤ p ≤ d − 1, but p ≠ d − 2; and E_{1}^{2(d−2),∗} = H^{∗}(BU(1)) ⊕ H^{∗}(BH^{ss}). Finally, we compute E_{1}^{2(h−1),∗} = H_{G}^{∗}(X_{h}). Set

Y_{h} = {B ∈ X_{h} : B = x^{h} · B′ = x^{h}(y^{h} + a_{2}x^{2}y^{h−2} + ⋯ + a_{h}x^{h}); and B′ is not an h-power}.

The stabilizer subgroup H of Y_{h} is the group of diagonal matrices of the form diag(α_{1}, α_{2}) with α_{1}^{h}α_{2}^{h} = 1. One can verify that X_{h} = G ×_{H} Y_{h}. Moreover, B′ is not a h-power if and only if (a_{2}, …, a_{h}) ≠ (0, …, 0). Hence Y_{h} is ℂ^{h−1} \ {0} and the action of H is a diagonal torus action (modulo a finite group). In particular, E_{1}^{2(h−1),∗} = H_{G}^{∗}(X_{h}) equals the cohomology ring of a weighted projective space of dimension h − 2, which is ℚ[t]/(t^{h−1}) (deg(t) = 2).

Proposition 4.7. The spectral sequence converges to H^{∗}(BG; ℚ) and it degenerates at E_{1}^{∗,∗}.

Proof. The Euler classes of the strata are not zero-divisors except for X_{h}. So we need the following local version of the Atiyah-Bott argument:

Lemma 4.8. Suppose that {X_{i}} is a G-equivariant stratification of V and the equivariant normal Euler class of X_{i} is not a zero-divisor if codim(X_{i}) > c. Then all differentials of the corresponding spectral sequence E_{r}^{p,q} starting or landing in the region p > c are zero.

Proof of Lemma. Let X be the union of X_{i} with codim(X_{i}) > c. Then the Lemma is equivalent with the statement that H_{G}^{∗}(V, V \ X) → H_{G}^{∗}(V) is injective, since {E_{r}^{p,q} : p > c} converges to H_{G}^{∗}(V, V \ X). Injectivity can be proved by adding the X_{i}’s one by one, and noticing that the composition

H_{G}^{n−2c}(X) ≅ H_{G}^{n}(U ∪ X_{i}, U) → H_{G}^{n}(U ∪ X_{i}) → H_{G}^{n}(X_{i})

is multiplication with the equivariant normal Euler class of the stratum X_{i} (where U is an open subset of V in which X_{i} is closed).

For the convenience of the reader we show how one determines the equivariant Euler class of o. Fix an element, say x^{h}y^{h} on o, let H^{ss} be its stabilizer, consider an H^{ss} invariant normal slice N at x^{h}y^{h}. In fact, for N one can take the vector space spanned by x^{i}y^{d−i}, where 0 ≤ i ≤ d, but i ∉ {h − 1, h, h + 1}. H^{ss} acts on N, and our goal is the computation of the Euler class e^{ss} ∈ H^{∗}(BH^{ss}) of EH^{ss} ×_{H^{ss}} N → BH^{ss}. Consider now the subgroup U(1) of H^{ss} (see 4.4). The Euler class e ∈ H^{∗}(BU(1)) = ℚ[t] of EH^{ss} ×_{U(1)} N → BU(1) can be computed as follows. The eigenvalues of diag(α, ᾱ) ∈ U(1) on N are (α^{d}, α^{d−2}, …, α^{4}, α^{−4}, …, α^{−d}), hence e = (dt)((d − 2)t) ⋯ (4t)(−4t) ⋯ (−dt) = mt^{d−2} for some m ≠ 0. Since d is even, this is in the invariants part H^{∗}(BH^{ss}) = ℚ[t^{2}] and can be identified in this ring by e^{ss}. Hence e^{ss} ≠ 0.

This type of argument is not working for the stratum X_{h} (since the stabilizer of its points are finite, and also H_{G}^{∗}(X_{h}) has zero divisors).

In order to show that the differentials d_{2h−2}^{0,q} (q odd and 2h − 3 ≤ q ≤ 4h − 7) of the spectral sequence are trivial, we consider another spectral sequence associated with only two strata, namely with X^{s} and X_{h}. The differential d_{2h−2}^{0,q} in the two spectral sequences coincides. If we compare them by the natural maps, then we get the exact sequence

0 → I′ → H_{G}^{∗}(V_{d}) →^{τ} H_{G}^{∗}(X^{s} ∪ X_{h})

where the ideal I′ is generated by all the columns E_{1}^{>2h−2,∗}. In E_{∞}^{2(h−1),2(j−1)} we can find special elements, those represented by the Thom polynomials Tp_{j} ∈ H_{G}^{∗}(V_{d}) associated with the partitions (1^{h−j}, j, h), where 0 < j < h. Hence, d_{2h−2}^{0,2j+2h−5} = 0 if τ(Tp_{j}) ≠ 0, or equivalently, if Tp_{j} ∉ I′. Notice that the graded ideal I′ and the graded ideal I considered in 4.2 and 4.3 are the same in the relevant degrees, hence it is enough to verify that Tp_{j} ∉ I for any j. But in 4.3 we verified that I = (Tp_{λ_{1}}, Tp_{λ_{2}}). Hence, we need to prove:

(1)  Tp_{j} ∉ (Tp_{λ_{1}}, Tp_{λ_{2}}).

Set

Π := ∏_{l=0}^{h−1}( lv + (d − l)u) and L = (h + 1)v + (h − 1)u.

From 3.7(1) one gets Tp_{λ_{1}} = hc_{1} · ∂(Π) and Tp_{λ_{2}} = hc_{1} · ∂(ΠL). In particular, by 3.9(3), Tp_{λ_{2}} = hL^{∗}c_{1} · ∂(Π) − 2hc_{1}Π, hence

(2)  (Tp_{λ_{1}}, Tp_{λ_{2}}) = (c_{1} · ∂(Π), c_{1} · Π).

Assume that (1) is not true and we have Tp_{j} = Ac_{1} · ∂(Π) + Bc_{1}Π. Since the degrees of Tp_{j} and Π are h + j − 2 and h respectively, the degree of Ac_{1} is j − 1. From 3.8 and 3.9(3), Tp_{j} = ∂(Π · D_{j}) = D_{j}^{∗} · ∂(Π) + Π · ∂(D_{j}). This means that

(3)  Π(∂(D_{j}) − Bc_{1}) = ∂(Π)(Ac_{1} − D_{j}^{∗}).

But it is easy to verify that gcd(Π, ∂(Π)) = 1. Indeed, if F |gcd(Π, ∂(Π)), then also F |(u − v)∂(Π) = Π − Π^{∗}, hence F |Π^{∗} as well. But gcd(Π, Π^{∗}) = 1.

This fact together with (3) show that Π|Ac_{1} − D_{j}^{∗}, but deg(Ac_{1} − D_{j}^{∗}) = j − 1 < deg Π, hence Ac_{1} = D_{j}^{∗}. In particular, c_{1}|D_{j}^{∗}, or u + v|D_{j}. But this leads to a contradiction. Indeed, analyzing in 3.8 the expression of (u − v)D_{j}, one sees that the first product is divisible by u + v (take l = h) but the second is not. Hence, (1) is true.

By similar argument as in the case of H_{G}^{∗}(X^{ss}), for H_{G}^{∗}(X^{s}) = H^{∗}(X^{s}/G) one gets:

Corollary 4.9. H^{odd}(X^{s}/G, ℚ) = 0, and the Poincaré series of H^{∗}(X^{s}/G) is the polynomial P (t) introduced in 4.4(1).

Let I′′ be the ideal in H^{∗}(BG) = ℚ[c_{1}, c_{2}] generated by the columns E_{1}^{>0,∗}. Then one has the ring isomorphism H^{∗}(X^{s}/G) = ℚ[c_{1}, c_{2}]/I′′. Now we will consider two special elements of I′′, namely the Thom polynomials Tp_{λ_{0}} and Tp_{λ′_{0}}, where λ_{0} = (1^{h}, h) and λ′_{0} = (1^{h−2}, 2, h).

Their degrees are h − 1 an h respectively. We will verify now that they are relative prime. Indeed, using the above notations, Tp_{λ_{0}} = ∂(Π) (from 3.7). Moreover, by 3.8 and 3.9(3) one has Tp_{λ′_{0}} = h(h−1)∂((u+3v)Π) = h(h−1)[(v +3u)∂Π−2Π]. In particular, gcd(Tp_{λ_{0}}, Tp_{λ′_{0}}) = gcd(Π, ∂Π) which is 1 by the proof of 4.7. Then the usual Poincaré polynomial argument shows I′′ = (Tp_{λ_{0}}, Tp_{λ′_{0}}) = (∂Π, Π).

This can be compared with (2) from the proof of 4.7: (Tp_{λ_{1}}, Tp_{λ_{2}}) = (c_{1}∂Π, c_{1}Π) (fact which can be deduced from 3.9(1) as well). Hence we proved:

Theorem 4.10. Assume d = 2h and set Π := ∏_{l=0}^{h−1}(lv + (d − l)u). Then:

H^{∗}(X^{s}/G; ℚ) = ℚ[c_{1}, c_{2}]/(Tp_{λ_{0}}, Tp_{λ′_{0}}) = ℚ[c_{1}, c_{2}]/(Π, ∂Π),

H^{∗}(X^{ss}/G, ℚ) = ℚ⟨1⟩ ⊕ \frac{c_{1}ℚ[c_{1}, c_{2}]}{(c_{1}Π, c_{1}∂Π)},

and the restriction map H^{∗}(X^{ss}/G) → H^{∗}(X^{s}/G) is induced by the identity of ℚ[c_{1}, c_{2}].

Discussion 4.11. The cohomology ring of the link. Denote by L^{ss} the link of the unique semisimple point p^{ss} in X^{ss}/G (i.e. L^{ss} = ρ^{−1}(ϵ), where ρ : X^{ss}/G → [0, ∞) is a real analytic map with ρ^{−1}(0) = {p^{ss}} and ϵ is sufficiently small). Write CL^{ss} for the real cone over it (i.e. CL^{ss} = [0, 1] × L^{ss}/{0} × L^{ss}). Then H^{∗}(CL^{ss}, L^{ss}) = H^{∗}(X^{ss}/G, X^{s}/G). Hence H^{∗}(L^{ss}) is completely determined by the restriction morphism from 4.10. In fact, L^{ss} is a rational homological manifold of real dimension 4h − 7 (with Poincaré duality). [This can also be proved as follows: The geometric quotient of the set of ordered d-points of ℙ^{1} is smooth, and one has only finitely many ordered semisimple points. Hence, L^{ss} is the quotient by a finite permutation group of a smooth (4h−7)–dimensional link]. 4.10, this duality and a computation give:

Theorem 4.12. H^{∗}(L^{ss}, ℚ) can be generated by two elements, c_{2} of degree 4 and g (the Poincaré dual of c_{2}^{[h/2]−1}) of degree 4h − 4[h/2] − 3 with relations c_{2}^{[h/2]} = 0 and g^{2} = 0. (Notice that all the Betti numbers are 0 or 1.)

Remark 4.13. 4.10 implies the following: the cohomology ring of the quasi-projective variety X^{s}/G of (complex) dimension d−3 shares the Poincaré duality properties of a smooth projective variety of dimension d − 4. In fact, cohomologically (over ℚ), X^{s}/G behaves like a line bundle L with Chern class c_{1} over a smooth projective variety M with cohomology ℚ[c_{1}, c_{2}]/(∂Π, Π); and X^{ss}/G behaves like the Thom space of this line bundle (or equivalently, the complex cone over M associated with L). In particular, L^{ss} has the cohomology of the S^{1}-bundle of L.

Remark 4.14. Assume that d = 2h + 1 is odd. It is tempting to compare the moduli space X^{s}/G with the (possibly weighted) Grassmanian Gr_{2}ℂ^{h+1} because the presentation of their cohomology rings have the same structure ℚ[c_{1}, c_{2}]/(∂p_{1}, ∂p_{2}) (where deg p_{1} = h+1 and deg p_{2} = h + 2), and they share the same Betti numbers. Indeed for the Grassmanian we can take p_{1} = u^{h+1}, p_{2} = u^{h+2}. In fact, this analogy can be continued: in both cases the set of relations are guided by some nice generating function, as follows. Set Π_{0} := 1 and Π_{j} := ∏_{l=0}^{j−1}(lv + (d − l)u), and consider the generating function

G(q) = ∑_{j≥0} G_{j}q^{j} := ∑_{j≥0} Π_{j}q^{j}/j! = [1 + (u − v)q]^{du/(u−v)} ∈ ℚ[u, v][[q]].

Then H^{∗}(X^{s}/G) = ℚ[c_{1}, c_{2}]/I, where I is generated by ∂G_{j}, j > h.

In the Grassmanian case the same fact is true with G(q) = 1 + uq + u^{2}q^{2} + ⋯ = 1/(1 − uq). However, easy computation shows that, as graded rings, these cohomology rings are not isomorphic (unless for small d’s).

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Department of Analysis, Eotvos University, Budapest

  E-mail address: lfeher@math-inst.hu

Department of Mathematics, The Ohio State University

  E-mail address: nemethi@math.ohio-state.edu

Department of Mathematics, The Ohio State University

  E-mail address: rimanyi@math.ohio-state.edu