How to calculate homology groups of spaces of nonsingular algebraic projective hypersurfaces

V. A. Vassiliev^{∗}

Abstract

A general method of computing cohomology groups of the space of nonsingular algebraic hypersurfaces of degree d in CP^{n} is described. Using this method, rational cohomology groups of such spaces with n = 2, d ≤ 4 and n = 3 = d are calculated.

1 Introduction

I describe a method of calculating the cohomology groups of spaces of nonsingular algebraic hypersurfaces of fixed degree in CP^{n}: a complex analog of the rigid homotopy classification of algebraic hypersurfaces in RP^{n}.

This method is essentially the one described in [12], [14]; it consists of two main parts. The first one is due to Arnold, who remarked [1] that often it is convenient to replace the calculation of cohomology groups of spaces of nonsingular objects by that of the (Alexander dual) homology groups of corresponding discriminant spaces of singular objects; see also [2]. The second one is the techniques of conical resolutions and topological order complexes, which is a continuous analog of the combinatorial inclusion-exclusion formula and is adopted very well to the calculation of the latter homology groups. This construction was proposed in [9] and generalized in [11], [12], [14].

The version of this method, applied to spaces of nonsingular projective hypersurfaces, is described in § 3. In the preceding § 2 we demonstrate all main ingredients of it, considering the simplest problem of this series: we calculate there the (wellknown) cohomology groups of the space of regular quadrics in CP^{2}. In § 4 we do the same for the real cohomology groups of the space of nonsingular cubics in CP^{2} (which probably also can be calculated by more standard methods, see remark after Theorem 2). In § 5, § 6 and § 7 we present similar calculations respectively for spaces of nonsingular quartics in CP^{2}, nonsingular cubics in CP^{3}, and nondegenerate quadratic vector fields in C^{3}.

∗supported in part by INTAS (project ????) and RFBR (project ????)

This work is inspired by the Arnold’s problem 1970-13 on the topology of the space of nonsingular cubics, see [3].

1.1 Stating the problem and the first reduction.

Let Π_{d,n} ≃ C^{D(d,n)} be the space of all homogeneous polynomials C^{n+1} → C^{1} of degree d, Σ_{d,n} ⊂ Π_{d,n} the set of forms, having singular points outside the origin, and N_{d,n} ⊂ CP^{D(d,n)−1} the projectivization of Π_{d,n} \ Σ_{d,n}, i.e. the set of nonsingular algebraic hypersurfaces of degree d in CP^{n}. Σ_{d,n} is a conical nonempty algebraic hypersurface in Π_{d,n}, therefore Π_{d,n} \Σ_{d,n} is diffeomorphic to N_{d,n} ×C^{∗}, in particular the homology groups of spaces N_{d,n} and Π_{d,n} \ Σ_{d,n} are related via the Künneth formula.

By the Alexander duality theorem, for i > 0 we have

H̃^{i}(Π_{d,n} \ Σ_{d,n}) ≃ H̄_{2D(d,n)−i−1}(Σ_{d,n}),  (1)

where H̄_{∗} is the notation for the Borel–Moore homology, i.e. the homology of the one-point compactification reduced modulo the added point.

2 The trivial example: nonsingular quadrics in CP^{2}

In this section we assume that n = 2 and denote the spaces Π_{d,2}, Σ_{d,2} and N_{d,2} simply by Π_{d}, Σ_{d} and N_{d}.

Proposition 1. Only the following groups H^{i}(Π_{2} \ Σ_{2}) are nontrivial: H^{0} ≃ H^{1} ≃ H^{5} ≃ H^{6} ≃ Z, H^{3} ≃ H^{4} ≃ Z_{2}. In particular, only the following groups H^{i}(N_{2}) are nontrivial: H^{0} ≃ H^{5} ≃ Z, H^{3} ≃ Z_{2}.

This statement is not new: indeed, it is easy to see that the space Π_{2} \ Σ_{2} is homotopy equivalent to the Lagrangian Grassmannian manifold U(3)/O(3), whose cohomology groups were studied in [4] and [6], see e.g. [10]. We give here another calculation, demonstrating our general method.

This calculation is based on the classification of subsets in CP^{2}, which can be the singular sets of homogeneous polynomials of degree 2 in C^{3}. There are exactly the following such sets:

A) any point x ∈ CP^{2},

B) any line CP^{1} ⊂ CP^{2} (such lines are parametrized by the points of the dual projective space CP^{2∨});

C) the entire CP^{2}.

For any set of type A), B) or C), the corresponding set of quadratic forms C^{3} → C, having singularities at all points of this set (and maybe somewhere else) is isomorphic, respectively, to C^{3}, C^{1} or C^{0}.

The main topological tool of the calculation is the following one.

2.1 The topological order complex Λ_{2}

Take the spaces CP^{2}, CP^{2∨} and {•} of sets of types A) B) and C), and consider the join CP^{2} ∗ CP^{2∨} ∗ {•} of these spaces, i.e., roughly speaking, the union of all simplices (of dimensions 0, 1 and 2), whose vertices correspond to their points.

Such a simplex is called coherent, if all subsets in CP^{2}, corresponding to its vertices, are incident to one another.

The topological order complex Λ_{2} is defined as the union of all coherent simplices, with topology induced from that of the join.

To any set K of the form A), B) or C) there corresponds a subset Λ_{2}(K) ⊂ Λ_{2} : the union of simplices, all whose vertices are subordinate by inclusion to the point {K}. Obviously this subset is a cone with the vertex {K}: for K of type A) (respectively, B), respectively, C)) it is equal to {K} itself (respectively, to the cone over the topological space K ≃ CP^{1}, respectively, to entire Λ_{2}). By Λ^{∘}_{2}(K) we denote this cone with its base removed.

2.2 Conical resolution

The conical resolution σ_{2} of Σ_{2} is defined as a subset in the direct product Π_{2} × Λ_{2}. Namely, let K ⊂ CP^{2} be any possible set of type A), B) or C). Let L(K) be the set of all quadratic forms C^{3} → C, having singular points at all points of K (and maybe somewhere else); this always is a vector subspace in Π_{2}. Then the space σ_{2}(K) ⊂ Π_{2} × Λ_{2} is defined as the direct product L(K) × Λ_{2}(K), and the desired conical resolution σ_{2} as the union of all possible sets σ_{2}(K).

Proposition 2 (see [12], [14]). The obvious projection σ_{2} → Σ_{2} is a proper map, and the induced map of one-point compactifications of these spaces is a homotopy equivalence. In particular this projection induces the isomorphism H̄_{∗}(σ_{2}) ≃ H̄_{∗}(Σ_{2}). □

The space σ_{2} admits a natural filtration F_{1} ⊂ F_{2} ⊂ F_{3}: the subspace F_{1} (respectively, F_{2}, respectively, F_{3}) of σ_{2} is the union of sets σ_{2}(K) such that the corresponding spaces K are of type A) only (respectively, of type A) or B), respectively, of type A) or B) or C), i.e. F_{3} = σ_{2}).

Similarly, we consider the filtration {Φ_{i}} of the space Λ_{2} : the subspace Φ_{1} (respectively, Φ_{2}, respectively, Φ_{3}) of Λ_{2} is the union of sets Λ_{2}(K) such that the corresponding spaces K are of type A) only (respectively, of type A) or B), respectively, of type A) or B) or C), i.e. Φ_{3} = σ_{2}).

In particular, the space F_{i} \ F_{i−1} is the space of a complex vector bundle over Φ_{i}\Φ_{i−1}, the dimension of this bundle is equal to 3 for i = 1, to 1 for i = 2 and to 0 for i = 3. Thus the Borel–Moore homology groups of spaces F_{i} \ F_{i−1} and Φ_{i}\Φ_{i−1} are related by the Thom isomorphism.

Consider the spectral sequence generated by the filtration {F_{i}} and converging to the group H̄_{∗}(σ_{2}). By definition its term E_{p,q}^{1} is equal to H̄_{p+q}(F_{p} \ F_{p−1}).

[figure: q; 9; 8; 7; 6; 5; 4; 3; Z; Z; Z; Z; Z; Z; Z; 1; 2; 3; p; q; 9; 8; 7; 6; 5; 4; 3; Z; Z; Z; 1; 2; 3; p; Z_{2}; Z_{2}]

Figure 1: Terms E^{1} and E^{2} of the spectral sequence for d = n = 2.

Lemma 1. All nontrivial terms E_{p,q}^{1} of this spectral sequence are as follows (see fig. 1 left): E_{1,q}^{1} = Z for q = 5, 7, 9; E_{2,q}^{1} = Z for q = 3, 5, 7; E_{3,q}^{1} = Z for q = 5.

Proof. 1. F_{1} is the space of a fiber bundle over CP^{2} with fiber C^{3}, this proves the statement concerning E_{1,q}^{1}.

2. F_{2} \ F_{1} is fibered over the space CP^{2∨} of all complex lines in CP^{2}. The fiber over such a line K is the direct product of the space L(K) ≃ C^{1} and the space Λ_{2}(K)\Φ_{1}. But Λ_{2}(K) is the cone over the set of all points of K ∼ CP^{1}, and the base of this cone belongs to Φ_{1}. Therefore Λ_{2}(K) \ Φ_{1} is an oriented 3-dimensional open disc, thus by the Thom isomorphism H̄_{∗}(F_{2}\F_{1}) ≃ H̄_{∗−5}(CP^{2∨}), and the statement concerning E_{2,∗}^{1} is proved.

3. F_{3} \ F_{2} ≡ Φ_{3} \ Φ_{2}. Φ_{2} is the subset of the join CP^{2} ∗ CP^{2∨}, consisting of all segments, spanning such pairs (x ∈ CP^{2}, K ∈ CP^{2∨}), that x ∈ K. In [11] it was proved that this subset is homeomorphic to S^{7}. Φ_{3} is the cone over Φ_{2}, hence F_{3} \F_{2} is an open 8-dimensional disc, and the statement concerning E_{3,∗}^{1} also is proved.

Proposition 3. The differential d^{1} of our spectral sequence maps a generator of the group E_{3,5}^{1} into twice a generator of E_{2,5}^{1}, maps a generator of E_{2,7}^{1} into twice a generator of E_{1,7}^{1}, and acts trivially on all other groups E_{p,q}^{1}. In particular all nontrivial groups E_{p,q}^{2} are as shown in fig. 1 right and our spectral sequence degenerates at this step: E^{2} ≡ E^{∞}.

Proposition 1 follows immediately from this one, Lemma 1 and the Alexander isomorphism (1).

Proof of Proposition 3. 1. The space F_{2}\F_{1} can be considered as the space of a fiber bundle with fiber C^{1}, whose base Φ_{2}\Φ_{1} is fibered over CP^{2∨} with fiber equal to an open 3-dimensional disc. The image of the map d^{1} : E_{3,5}^{1} → E_{2,5}^{1} is equal to the fundamental cycle (with closed supports) of the base Φ_{2} \Φ_{1} of this bundle. The homomorphism

H̄_{7}(Φ_{2}\Φ_{1}) → H̄_{7}(F_{2}\F_{1}) → H̄_{5}(Φ_{2}\Φ_{1}),  (2)

defined as the composition of the identical embedding and the Thom isomorphism, maps this fundamental cycle into a homology class, Poincaré dual to the Euler class of the complex line bundle, whose fiber over the points of the open cone Λ^{∘}_{2}(K), K ∈ CP^{2∨}, is equal to the space L(K). This space consists of all quadratic functions of the form l^{2}, where l is an arbitrary linear form vanishing on the line K. It is easy to calculate that this Euler class is equal to twice the generator of the group H̄_{5}(Φ_{2}\Φ_{1}), and the first statement of Proposition 3 is proved.

2. The homomorphism d^{1} : E_{2,7}^{1} → E_{1,7}^{1} maps the generator of the group E_{2,7}^{1} into the fundamental cycle of the algebraic subset ω ⊂ F_{1}, consisting of all pairs of the form (a point u ∈ CP^{2}, a quadratic function l^{2}), such that l is a linear form with l(u) = 0. This subset is fibered over CP^{2} with a standard quadratic cone {αβ = γ^{2}} ⊂ C^{3} for a fiber. To calculate the homology class of this cycle, it is sufficient to calculate its intersection index with the generator of the group H_{2}(F_{1}). This generator can be realized as follows: we take an arbitrary line CP^{1} ⊂ CP^{2}, say the one given by the equation x = 0, and consider any section of the restriction on it of the fibration F_{1} → CP^{2}. For such a section we can take the one, whose value over the point (0 : 1 : λ) ∈ CP^{1} is the quadratic form \frac{x(z−λy)}{1+|λ|^{2}}. This section intersects the above-described set ω at exactly one point, corresponding to λ = ∞. This intersection is of multiplicity 2, hence the fundamental cycle of this set ω is equal to twice the generator of H̄_{8}(F_{1}).

It is now obvious that d^{1} cannot be nontrivial on any other group E_{p,q}^{1}, (p, q) ≠ (3, 5), (2, 7), and Proposition 3 is completely proved.

3 The general spectral sequence

Here we extend the above-described construction to the case of hypersurfaces of an arbitrary degree d. The spaces of sets of types A), B) and C) will be formalized in terms of the Hilbert-scheme topology.

3.1 Conical resolution of the discriminant: preliminary version.

Set D ≡ D(d, n) = dim Π_{d,n}. Consider all Grassmann manifolds G_{i}(Π_{d,n}), i = 1, …, D, whose points are the complex subspaces of codimension i in Π_{d,n}. Suppose that the subset K ⊂ CP^{n} is the set of singular points of a certain homogeneous function C^{n+1} → C^{1}, and denote by L(K) the subspace in Π_{d,n} consisting of all functions having singular points everywhere in K (and maybe somewhere else). Obviously this is a vector subspace in Π_{d,n}. Let Ω_{i} ⊂ G_{i}(Π_{d,n}) be the set of all subspaces of the form L(K). Let Ω̄_{i} be the closure of Ω_{i} in G_{i}(Π_{d,n}).

Consider the join G_{1}(Π_{d,n}) ∗ ⋯ ∗ G_{D−1}(Π_{d,n}) ∗ G_{D}(Π_{d,n}) and the subset in it, consisting of all coherent simplices of any dimensions, i.e. of such simplices, that a) all their vertices belong to subspaces Ω̄_{i}, and b) the collection of subspaces in Π_{d,n}, corresponding to all vertices of the simplex, forms a flag.

The union of all such simplices will be denoted by Λ̃_{d,n}. The leading vertex of a coherent simplex is the one lying in the space Ω̄_{i} with the greatest value of i. For any point {L} ∈ Ω̄_{i}, the set Λ̃_{d,n}(L) is the union of coherent simplices, whose leading vertices coincide with L. Obviously this is a cone with vertex {L}. In particular, Λ̃_{d,n} = Λ̃_{d,n}({Π_{d,n}}).

The subset σ̃_{d,n}(L) ⊂ Π_{d,n} × Λ̃_{d,n} is defined as the direct product L × Λ̃_{d,n}(L). The conical resolution σ̃_{d,n} of Σ_{d,n} is defined as the union of all sets σ̃_{d,n}(L) over all points {L} of all Ω̄_{i}, i = 1, …, D.

Proposition 4. The obvious projection σ̃_{d,n} → Σ_{d,n} is a proper map, and the induced map of one-point compactifications of these spaces is a homotopy equivalence. In particular it induces an isomorphism H̄_{∗}(σ̃_{d,n}) ≃ H̄_{∗}(Σ_{d,n}).

The space σ̃_{d,n} admits a natural increasing filtration: its term F̃_{i} ⊂ σ̃_{d,n} is the union of sets σ̃_{d,n}(L) over all L ∈ Ω̄_{j}, j ≤ i.

Simultaneously we consider the increasing filtration {Φ̃_{i}} on the space Λ̃_{d,n} : its term Φ̃_{i} is the union of sets Λ̃_{d,n}(L) such that codim_{C}L ≤ i. In particular, the space F̃_{i} \ F̃_{i−1} is the space of a complex (D − i)-dimensional vector bundle over Φ̃_{i} \ Φ̃_{i−1}, and their Borel–Moore homology groups are related by the Thom isomorphism.

The simplicial resolution of § 2 is the special case of this one, and its filtration F_{1} ⊂ F_{2} ⊂ F_{3} is obtained from our present filtration {F̃_{i}} by renumbering its terms: F_{1} = F̃_{3} = F̃_{4}, F_{2} = F̃_{5}, F_{3} = F̃_{6}.

3.2 Reduced conical resolutions of discriminants.

There is another, more economical conical resolution of Σ_{d,n}, also coinciding with the previous one in the case d = 2 = n. Namely, we can reduce all ”nongeometrical” vertices L ∈ Ω̄_{i}\Ω_{i} and their subordinate simplices.

Definition. Given a point L ∈ Ω̄_{i}, its geometrization L̄ is the smallest plane of the form L(K) ∈ Ω_{j}, j ≤ i, containing L. In particular L̄ = L if L ∈ Ω_{i}.

Example. Suppose that d ≥ 3, u ∈ CP^{n}, and v(t), t ∈ C^{1}, is an one-parametric analytic family of points in CP^{n} such that v(t) = u ⇔ t = 0. Then for any t ≠ 0 the 2-point set K(t) ≡ (u, v(t)) defines a subspace L(t) ≡ L(K(t)) of codimension 2(n+1) in Π_{d,n}, while the limit position L(0) of these subspaces when t → 0 belongs to Ω̄_{2(n+1)}\Ω_{2(n+1)}. Then \overline{L(0)} = L(\{u\}) ∈ Ω_{n+1}.

The segment in Λ̃_{d,n}, joining these two points L(0) and L({u}), is homotopy equivalent to its endpoint L({u}), therefore we can contract it to this point, and to extend this retraction by linearity to all coherent simplices, containing this segment. Doing the same in all similar situations, we reduce the complex Λ̃_{d,n} to the (properly topologized) union Λ_{d,n} of all coherent simplices, whose vertices correspond to the points of spaces Ω_{i} only, but not of Ω̄_{i}\Ω_{i}. Then we lift this reducing map to the space σ̃_{d,n} and get the desired space σ_{d,n}, with which we will work all the time. This general reduction map is constructed as follows.

Given a point L ∈ Ω_{j}, denote by W (L) the union of all coherent simplices in Λ̃_{d,n} such that for all their vertices L′ their geometrizations \overline{L′} coincide with L. Obviously W (L) is a compact and contractible subspace of Λ̃_{d,n}: it is a cone with vertex {L}.

The reducing map

red : Λ̃_{d,n} → Λ_{d,n}  (3)

is defined as follows. For any i and any L ∈ Ω̄_{i} we map the point L to L̄ and extend this map by linearity to all coherent simplices; the image of a coherent simplex under this map obviously is again coherent. Then we call equivalent all points of Λ̃_{d,n} sent by this extended map into one and the same point. The space Λ_{d,n} is defined as the quotient space of Λ̃_{d,n} by this equivalence relation.

This factorization map defines obviously a map Λ̃_{d,n} × Π_{d,n} → Λ_{d,n} × Π_{d,n}, and hence also a map

Red : σ̃_{d,n} → σ_{d,n}  (4)

of the resolved discriminant σ̃_{d,n} ⊂ Λ̃_{d,n} × Π_{d,n} onto some space σ_{d,n} ⊂ Λ_{d,n} × Π_{d,n}.

Proposition 5. The map (4) induces a homotopy equivalence of one-point compactifications of spaces σ̃_{d,n} and σ_{d,n} and factorizes the map π : σ̃_{d,n} → Σ (i.e., there is a proper map π̃ : σ_{d,n} → Σ such that π = π̃ ∘ Red).

This follows immediately from the construction. □

For any L ∈ Ω_{i} denote by Λ_{d,n}(L) the image of Λ̃_{d,n}(L) under the reducing map (3).

Example. If K is a set of r < ∞ points, and d is sufficiently large with respect to r, then both spaces Λ̃_{d,n}(L(K)) and Λ_{d,n}(L(K)) are naturally homeomorphic to a (r − 1)-dimensional simplex, and the restriction of the map red onto Λ̃_{d,n}(K(L)) is a homeomorphism between them.

Definitions and notation. For any set L we denote by ∂Λ_{d,n}(L) the link of the order complex Λ_{d,n}(L), i.e. the union of all its coherent simplices not containing its vertex {L}, and denote by Λ^{∘}_{d,n} (L) the open cone Λ_{d,n}(L)\∂Λ_{d,n}(L). Their homology groups are obviously related by the boundary isomorphism

H̄_{∗}(Λ^{∘}_{d,n} (L)) ≅ H_{∗}(Λ_{d,n}(L), ∂Λ_{d,n}(L)) →^{∼} H_{∗−1}(∂Λ_{d,n}(L), pt).  (5)

For any subset K ⊂ CP^{n} the complex Λ_{d,n}(L(K)) will be usually denoted simply by Λ_{d,n}(K).

3.3 Filtrations and spectral sequences.

The space Λ̃_{d,n} admits a natural increasing filtration Φ̃_{1} ⊂ ⋯ ⊂ Φ̃_{D} ≡ Λ̃_{d,n}. Namely, the filtration of any interior point of a coherent simplex Δ ⊂ Λ̃_{d,n} is equal to the codimension i of the plane L(Δ) ∈ Ω̄_{i} corresponding to the leading vertex of this simplex. Hence, a similar filtration is induced also on the quotient space Λ_{d,n}: the filtration of the equivalence class x ∈ Λ_{d,n} is equal to the smallest value of filtrations of points in Λ̃_{d,n}, constituting x. The obvious projections σ̃_{d,n} → Λ̃_{d,n}, σ_{d,n} → Λ_{d,n} induce also filtrations on spaces σ̃_{d,n} and σ_{d,n}. By construction, the term F̃_{i} \ F̃_{i−1} of this filtration on σ̃_{d,n} (respectively, on σ_{d,n}) is the space of a (D − i)-dimensional complex vector bundle over the term Φ̃_{i}\Φ̃_{i−1} of the filtration on Λ̃_{d,n} (respectively, on Λ_{d,n}).

In fact, we will use a shortened version of this filtration, removing all nonincreasing terms by the following inductive rule: its term F_{1} is the first nonempty set F̃_{j}, and its term F_{i} is the first term F̃_{j} of the former filtration which is strictly greater than F_{i−1}.

This filtration defines a spectral sequence, converging to the right-hand group in (1). Its term E_{p,q}^{1} is isomorphic to H̄_{p+q}(F_{p} \ F_{p−1}).

4 Non-singular forms of degree 3

Theorem 1. Only the following groups H^{i}(Π_{3,2} \ Σ_{3,2}, R) are nontrivial: H^{0} ≃ H^{1} ≃ H^{3} ≃ H^{4} ≃ H^{5} ≃ H^{6} ≃ H^{8} ≃ H^{9} ≃ R. In particular, only the following groups H^{i}(N_{3,2}, R) are nontrivial: H^{0} ≃ H^{3} ≃ H^{5} ≃ H^{8} ≃ R.

This theorem follows immediately from the following one.

Theorem 2. The term E^{1} of our spectral sequence (over R), converging to the group H̄_{∗}(Σ_{3,2}), looks as in fig. 2 (where empty cells denote trivial groups E_{p,q}^{1}). E^{∞} ≡ E^{1}, i.e. the unique possible differential d_{1} : E_{2,13}^{1} → E_{1,13}^{1} of this sequence is trivial.

Remark. Theorem 1 is not surprising: it states in fact that H^{∗}(Π_{3,2}\Σ_{3,2}, R) ≃ H^{∗}(P GL(CP^{2}), R). There is obvious projection of the space Π_{3,2}\Σ_{3,2} onto the modular space of elliptic curves. The group P GL(CP^{2}) acts transitively on all fibers of this projection, and the stationary groups of this action are finite for all fibers (although they are not the same: they ”jump” over the curve with additional symmetry). It seems likely that Theorem 1 could be proved by considering these subgroups and the Leray spectral sequence of this projection.

The proof of Theorem 2 occupies the rest of this section. It uses many times the following notions and facts.

[figure: p; q; 17; 16; 15; 14; 13; 12; 11; 10; 9; 8; 7; 6; R; R; R; R; R; R; R; 5_{(10)}; 4_{(9)}; 3_{(7)}; 2_{(6)}; 1_{(3)}]

Figure 2: spectral sequence for d = 3.

4.1 Main homological lemmas.

Definition. Given a topological space X, the k-th configuration space B(X, k) of X is the space of all subsets of cardinality k in X, supplied with the natural topology, see e.g. [12], [14]. The sign representation π_{1}(B(X, k)) → Aut(Z) maps the paths in B(X, k), defining odd (respectively, even) permutations of k points into multiplication by −1 (respectively, 1). The local system ±Z over B(X, k), is the one locally isomorphic to Z, but with monodromy representation equal to the sign representation of π_{1}(B(X, k)). The twisted Borel–Moore homology group H̄_{∗}(B(X, k), ±Z) is defined as the homology group of the complex of locally finite singular chains of B(X, k) with coefficients in the sheaf ±Z. Similarly we define local systems ±R ≡ ±Z ⊗ R and their homology groups. For any topological space X, X̄ denotes the one-point compactification of X.

Lemma 2. A. The group

H̄_{∗}(B(C^{n}, k), ±R),  (6)

n ≥ 1, is trivial for any k ≥ 2.

B. The group

H̄_{∗}(B(CP^{n}, k), ±R),  (7)

n ≥ 1, is isomorphic to H_{∗−k(k−1)}(G_{k}(C^{n+1}), R), where G_{k}(C^{n+1}) is the Grassmann manifold of k-dimensional subspaces in C^{n+1}, see [8].

In particular, the group (7) is trivial if k > n + 1.

Proof. Assertion A is well-known, see e.g. [12], [14].

Let us fix some complete flag • ⊂ CP^{1} ⊂ CP^{2} ⊂ ⋯ ⊂ CP^{n}. To any configuration of k points in CP^{n} associate its index a = (a_{0} ≤ a_{1} ≤ ⋯ ≤ a_{n}), a_{n} = k, where a_{i} is the number of its points lying in CP^{i}. For any such index a denote by B_{a} ⊂ B(CP^{n}, k) the union of all configurations having this index. It follows immediately from statement A, that H̄_{∗}(B_{a}, ±R) = 0 for any index a such that a_{i} − a_{i−1} > 1 for at least one i. Thus only the indices a, which are Schubert symbols of certain Schubert cells of G_{k}(C^{n+1}), can provide nontrivial groups H̄_{∗}(B_{a}, ±R). Moreover, for any Schubert symbol a the set B_{a} is diffeomorphic to the complex vector space of dimension ∑_{i=1}^{n} i · (a_{i} − a_{i−1}). This implies statement B.

Definition. For any finite-dimensional topological space X, its k-th self-join X^{∗k} is defined as follows: we embed X generically into the space R^{N} of a very large dimension and take the union of all (k − 1)-dimensional simplices spanning all ktuples of points in X. (The genericity of the embedding means that these simplices have only obvious intersections, namely, the intersection set of any two simplices is some their common face.)

Lemma 3. If k ≥ 2, then H_{∗}((S^{2})^{∗k}, R) ≡ 0 in all dimensions.

Proof. The space (S^{2})^{∗k} is obviously filtered by similar self-joins:

S^{1} ≡ (S^{2})^{∗1} ⊂ (S^{2})^{∗2} ⊂ ⋯ ⊂ (S^{2})^{∗(k−1)} ⊂ (S^{2})^{∗k}.  (8)

Consider the spectral sequence generated by this filtration. Any space ϕ_{i}\ϕ_{i−1} ≡

(S^{2})^{∗i}\(S^{2})^{∗(i−1)} of this filtration is fibered over B(S^{2}, i) with the fiber Δ^{∘}^{i−1}. This bundle is nontrivial, its fibers change their orientations exactly over the same loops, which act nontrivially on the local system ±R. Hence the first term of this sequence is given by the formula

E_{p,q}^{1} ≃ H̄_{p+q}((S^{2})^{∗p}\(S^{2})^{∗(p−1)}, R) ≃ H̄_{q+1}(B(CP^{1}, p), ±R).

By Lemma 2 there are exactly two such nontrivial cells E_{p,q}^{1}, namely E_{2,1}^{1} ≃ E_{1,1}^{1} ≃ R. The unique differential d^{1} of our spectral sequence, which can be nontrivial, acts between these cells; it is exactly the boundary homomorphism H_{3}((S^{2})^{∗2}, S^{2}; R) → H_{2}(S^{2}, R); It follows immediately from the shape of the generator of the former group (i.e., from that of the 2-dimensional Schubert cell B_{a}, a = (1, 2) in B(S^{2}, 2)) that this differential an isomorphism. This implies our Lemma.

Denote by B̃(CP^{2}, k) the subset in B(CP^{2}, k), consisting of generic configurations, i.e. of such that none three points do lie in the same line.

Lemma 4. There are isomorphisms

H̄_{∗}(B(CP^{2}, 3), ±R) ⟶^{∼} H̄_{∗}(B̃(CP^{2}, 3), ±R),  (9)

H̄_{∗}(B(CP^{2}, 4), ±R) ⟶^{∼} H̄_{∗}(B̃(CP^{2}, 4), ±R),  (10)

induced by identical embeddings. Namely, both groups (10) are trivial in all dimensions, and groups (9) are trivial in all dimensions other than 6 and are isomorphic to R in dimension 6.

Proof. The space B(CP^{2}, 3)\B̃(CP^{2}, 3) consists of 3-configurations lying on the same line. This space is obviously fibered over CP^{2∨} with fiber equal to B(CP^{1}, 3). Applying Lemma 2 B) to this fiber, we get the assertion about the groups (9). Similarly, the space B(CP^{2}, 4)\B̃(CP^{2}, 4) is the union of two pieces, consisting of 4configurations, some three (respectively all four) points of which lie in the same line. Both these pieces are fibered over CP^{2∨} with fibers B(CP^{1}, 3) × C^{2} (respectively, B(CP^{1}, 4)). By Lemma 2 B) both these pieces are ±R-acyclic, and Lemma 4 is proved.

4.2 Proof of Theorem 2.

There are exactly the following possible singular sets K ⊂ CP^{2} defined by homogeneous polynomials C^{3} → C^{1} of degree 3:

A) any point u ∈ CP^{2};

B) any pair of points in CP^{2}; the corresponding space L(K) consists of all 3forms splitting into the product of two polynomials of degree 1 and 2, vanishing at these two points;

C) any line in CP^{2} : the corresponding space L(K) consists of 3-forms vanishing on this line with multiplicity ≥ 2;

D) any triple of points in CP^{2} not lying on the same line, i.e. defining a point of the space B̃(CP^{2}, 3). The space L(K) in this case is 1-dimensional and consists of forms vanishing on any of three lines containing any two of our three points;

E) the entire CP^{2}.

The corresponding terms F_{i} \ F_{i−1} of our filtration and the corresponding terms of the spectral sequence look as follows. Any of them is the space of a fibered product of two fiber bundles, whose base is the corresponding space of all possible sets K (i.e., respectively, CP^{2}, B(CP^{2}, 2), CP^{2∨}, B̃(CP^{2}, 3) and {pt}). The fiber of the first (respectively, second) bundle over any point {K} of such a base is the space L(K) (respectively, the open cone Λ^{∘}_{3,2} (K)). The spaces of only second fiber bundles are exactly the terms Φ_{i}\Φ_{i−1} of the related filtration of the continuous order complex Λ_{3,2} of all possible sets K defined by homogeneous polynomials of order 3.

Let us describe explicitly all these terms and bundles.

A) The space F_{1} ≡ F̃_{3} is the space of a fiber bundle with fiber C^{7} over CP^{2}, this implies statement of Theorem 2 concerning the column p = 1^{1}.

1the fibers of the second bundle in this case are the points

B) The space F_{2}\F_{1} ≡ F̃_{6}\F̃_{5}. The fiber of the second bundle over a point (u, v) ∈ B(CP^{2}, 2) is the union of two segments, joining this point (u, v) with both points u and v ∈ CP^{2}, and not containing their boundary points u and v (which belong to F_{1}). Obviously such an union is homeomorphic to the open segment (−1, 1). Our bundle of such segments is not oriented: it changes the orientation simultaneously with the sheaf ±Z. Therefore the term E_{2,q}^{1} of our spectral sequence is given by the formula

E_{2,q}^{1} ≅ H̄_{2+q}(F_{2}\F_{1}, R) ≃ H̄_{2+q−8}(Φ_{2}\Φ_{1}, R) ≃ H̄_{2+q−9}(B(CP^{2}, 2), ±R). (11)

Thus the assertion of Theorem 2 concerning the column E_{2,q}^{1} of the main spectral sequence follows from Lemma 2.

C) Let K = CP^{1} ⊂ CP^{2}. Then the reduced order complex Λ_{3,2}(K) is homeomorphic to the cone with vertex {L(K)} ⊂ Ω_{7}, and base lying in Φ_{2}. By the construction, this base ∂Λ_{3,2}(K) is canonically homeomorphic to the self-join (CP^{1})^{∗2}, whose homology groups are trivial by Lemma 3.

The space F_{3}\F_{2} is fibered over CP^{2∨}; its fiber over a line {K} ∈ CP^{2∨} is the product of the space C^{3} and the space Λ_{3,2}(K)\Φ_{2}, i.e. an open cone with base ∂Λ_{3,2}(K). Thus the Borel–Moore homology group of this fiber is acyclic over R, this implies assertion of Theorem 2 concerning the column p = 3.

D) Φ_{4}\Φ_{3} is fibered over B̃(CP^{2}, 3); its fiber over a point (u, v, w) ⊂ CP^{2} is the open triangle Δ̇^{2}, whose vertices are identified with these points u, v and w. In particular,

H̄_{i}(F_{4}\F_{3}) ≡ H̄_{i−2}(Φ_{4}\Φ_{3}) ≃ H̄_{i−4}(B̃(CP^{2}, 3), ±Z).  (12)

By Lemmas 4 and 2 B), the group H_{j}(B̃(CP^{2}, 3), ±R) is isomorphic to R if j = 6 and is trivial for all j ≠ 6. This proves the assertion of Theorem 2 concerning the column p = 4.

E) Finally, the set F_{5}\F_{4} coincides with Φ_{5}\Φ_{4} ≡ Λ_{3,2}\Φ_{4}; it is an open cone over the reduced topological order complex Φ_{4}, so that H̄_{i}(F_{5}\F_{4}) ≃ H̃_{i−1}(Φ_{4}). Consider the spectral sequence e_{b,c}^{a} → H_{∗}(Φ_{4}, R), generated by the filtration {Φ_{i}}, i = 1, 2, 3, 4. Its term e_{i,c}^{1} is isomorphic to H̄_{i+c}(Φ_{i}\Φ_{i−1}, R). The homology groups of spaces (Φ_{i}\Φ_{i−1}) are related by the Thom isomorphism with these of corresponding spaces F_{i}\F_{i−1} (calculated in the previous steps A — D), hence the term e^{1} of our sequence looks as is shown in fig. 3.

Lemma 5. The differentials ∂^{1} : e_{2,1}^{1} → e_{1,1}^{1} and ∂^{1} : e_{2,3}^{1} → e_{1,3}^{1} of our spectral sequence are isomorphisms, as well as the map ∂^{2} : e_{4,4}^{2} → e_{2,5}^{2}. In particular, the term e^{∞} ≡ e^{3} of our spectral sequence consists of unique nontrivial cell e_{1,−1}^{3} ≃ R.

Proof. If some of these maps is not isomorphic, then there exists a nontrivial group H_{i−1}(Φ_{4}, R) ≃ H̄_{i}(F_{5}\F_{4}, R) with i < 9. It follows from the shape of terms E_{p,q}^{1} of the main spectral sequence with p ≤ 4 (determined in the previous steps) that the corresponding group E_{5,i−5}^{1} cannot disappear in the further steps of this

[figure: p; q; 8; 7; 6; 5; 4; 3; 2; 1; 0; −1; R; R; R; R; R; R; R; 4; 3; 2; 1]

Figure 3: spectral sequence for the term F_{5} \ F_{4}.

sequence and thus the group H̄_{i}(Σ_{3,2}) ≃ H^{19−i}(Π_{3,2}\Σ_{3,2}) also is nontrivial. This is impossible because Π_{3,2}\Σ_{3,2} is a 10-dimensional Stein manifold. □

Thus we have proved the assertion of Theorem 2 concerning the shape of the term E^{1} of the main spectral sequence.

Lemma 6. The homomorphism d^{1} : E_{2,13}^{1} → E_{1,13}^{1} is trivial.

Proof. It is clear from fig. 2, that the group E_{1,15} cannot disappear, hence H̄_{16}(Σ_{3,2}, R) ≃ H^{3}(Π_{3,2}\Σ_{3,2}, R) ∼ R. Π_{3,2}\Σ_{3,2} is a direct product of something and C^{∗}, therefore the Poincaré polynomial of H^{∗}(Π_{3,2}\Σ_{3,2}, R) is divisible by (1+t). Since H̄_{17}(Σ_{3,2}, R) ≃ H^{2}(Π_{3,2}\Σ_{3,2}, R) = 0, this implies that H̄_{15}(Σ_{3,2}, R) ≠ 0. But the unique cell E_{p,q}^{1} with p + q = 15 is E_{2,13}^{1}, and our lemma is proved.

It terminates the proof of Theorems 2 and 1.

5 Non-singular quartics in CP^{2}

Theorem 3. The Poincaré polynomial of the real cohomology group of the space N_{4,2} of nonsingular curves of degree 4 in CP^{2} is equal to (1 + t^{3})(1 + t^{5})(1 + t^{6}).

For the proof, consider the standard conical resolution of Σ_{4,2} described in § 3. To describe it explicitly, we must classify all singular sets in CP^{2} of polynomials C^{3} → C^{1} of degree 4.

Proposition 6. There are exactly the following possible singular sets in CP^{2} of homogeneous polynomials of degree 4 in C^{3} (in angular parentheses we indicate the complex dimension of the space of all polynomials having the singularity at an arbitrary fixed set of the corresponding class):

1. Any point in CP^{2} ⟨12⟩

2. Any pair of points ⟨9⟩

3. Any three point on the same line ⟨7⟩ [figure]

4. Any generic triple of points (i.e. not lying on the same line) ⟨6⟩

5. Any line CP^{1} ⊂ CP^{2} ⟨6⟩

6. Any three points on the same line plus a point not on the line ⟨4⟩ [figure]

7. Any generic quadruple of points (i.e., none three of them are on the same line; the corresponding polynomial necessarily splits into the product of two quadrics) ⟨3⟩

8. Any line CP^{1} ⊂ CP^{2} plus a point outside it ⟨3⟩

9. Any five points, four of which are generic, and the fifth is the intersection point of two lines spanning some pairs of first four points (the corresponding polynomial splits into the product of two linear functions and one quadric) ⟨2⟩[figure]

10. Six points, which are all the intersection points of some four lines in general

position ⟨1⟩ [figure]

11. Any nonsingular quadric in CP^{2} ⟨1⟩

12. Two lines in CP^{2} ⟨1⟩

13. Entire CP^{2}. ⟨0⟩ □

I have liberty to improve slightly the canonical filtration in the resolution space σ_{4,2} of the discriminant and in the underlying order complex Λ_{4,2} of singular sets. Namely, for any i = 1, …, 13, I denote by F_{i} (respectively, by Φ_{i}) the union of all sets σ_{4,2}(K) ⊂ σ_{4,2} (respectively, Λ_{4,2}(K) ⊂ Λ_{4,2}) where K is one of sets described in items 1, …, i of the previous proposition.

Proposition 7. The term E^{1} of the spectral sequence, generated by this filtration and converging to the group H̄_{∗}(σ_{4,2}, R), is as shown in Fig. 4. This spectral sequence degenerates at this term: E^{∞} ≡ E^{1}.

Proof. The shape of columns p = 1, 2 and 4 of E^{1} is justified in essentially the same way as that of the same columns of the spectral sequence presented in Fig. 2.

[figure: 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; q; p; R; R; R; R; R; R; R; 15; 14; 13; 12; 11; 10; 9; 8; 7; 6; 5; 4; 3; 2; 1; R; R; R; R; R; R; R; R]

Figure 4: Spectral sequence for the space of nonsingular homogeneous polynomials of degree 4 in C^{3} The triviality of columns p = 3 and p = 6 follows from Lemma 2 (n = 1, k = 3), and that of the column p = 7 from the same lemma and additionally Lemma 4. For any set K of type 5 (i.e., K = CP^{1}) the link ∂(Λ_{4,2}(K)) of the corresponding order complex is the space (CP^{1})^{∗3}, thus the triviality of the column p = 5 follows from Lemma 3. Similarly, for any set K of type 11 (i.e., K is a generic quadric in CP^{2}) the link ∂(Λ_{4,2}(K)) is the space K^{∗4}, and the triviality of the column p = 11 also follows from Lemma 3.

The space of configurations of type 9 is fibered over B(CP^{2∨}, 2) with fiber B(C^{1}, 2) × B(C^{1}, 2), and everything dyes already over this fiber. Thus the column p = 9 also is trivial.

In the case mentioned in the column p = 10 the term F_{10}\F_{9} is the space of a fiber bundle, whose base is the configuration space B̃(CP^{2∨}, 4) of all generic collections of four lines in CP^{2} (i.e., such that none three of them intersect at the same point), and the fiber is the direct product of a complex line and a 5-dimensional open simplex, whose 6 vertices correspond to all intersection points of these four lines. It is easy to check that this bundle of 5-simplices is orientable, thus H̄_{∗}(F_{10}\F_{9}, R) ≃ H̄_{∗−7}(B̃(CP^{2∨}, 4), R). Let us calculate the latter homology group. By the projective duality, B̃(CP^{2∨}, 4) is diffeomorphic to the space B̃(CP^{2}, 4) of all generic collections of 4 points, see e.g. Lemma 4. The similar space F̃(CP^{2}, 4) of ordered generic collections of points is obviously diffeomorphic to the group P GL(CP^{2}), whose real homology groups are well-known. It is easy to check that these homology groups do not disappear when we pass to the base of the 24-fold covering F̃(CP^{2}, 4) ⟶^{S(4)} B̃(CP^{2}, 4), and the shape of the column p = 10 follows.

The triviality of the column p = 8 follows from the following lemma.

Lemma 7. For any set K ⊂ CP^{2} of type 8, K = CP^{1} ⊔ ∗, the link ∂Λ_{4,2}(K) of the corresponding order complex Λ_{4,2}(K) is homeomorphic to the suspension Σ(∂Λ_{4,2}(CP^{1})) of the similar link for the set CP^{1} of type 5.

Proof. This link consists of two pieces, the first of which is the order complex Λ_{4,2}(CP^{1}) (obviously homeomorphic to the cone over its own link), and the second is the union of all order complexes Λ_{4,2}(κ), where κ is some subset of type 6 in K. All the latter order complexes are the tetrahedra, one of whose 4 vertices is the distinguished point ∗ = K\CP^{1}, and another three are arbitrary different points of CP^{1}. Thus the second piece also is homeomorphic to the cone over ∂Λ_{4,2} ∼ (S^{2})^{∗3}. The intersection of these two cones is their common base, and Lemma 7 is proved.

Now let K be of type 12, i.e. the union of two complex lines l_{1}, l_{2} ⊂ CP^{2}. The corresponding link ∂Λ_{4,2}(K) is covered by two diffeomorphic sets, A_{1} and A_{2}. Namely, A_{i}, i = 1, 2, is the union of all continuous order complexes Λ_{4,2}(κ), where κ is the set of type 8 consisting of the line l_{i} and a point of the other line K\l_{i}. The triviality of the column p = 12 follows now from the Mayer–Vietoris formula and the following lemma.

Lemma 8. All three complexes A_{1}, A_{2} and A_{1} ∩ A_{2} are acyclic (over R) in all positive dimensions.

Proof. Consider the following filtration of the set A_{1}:

Λ_{4,2}(l_{1}) ⊂ A_{1} ∩ Φ_{6} ⊂ A_{1} ∩ Φ_{8} ≡ A_{1}.

The set A_{1}\(A_{1} ∩ Φ_{6}) is a fiber bundle, whose base is the space (K\l_{1}) ∼ C^{1}, parametrizing all possible sets of type 8 containing l_{1}, and fiber Λ^{∘}_{4,2} (κ), where κ is such a set. By Lemma 7 the Borel–Moore homology group of this difference (or, which is the same, the relative group H_{∗}(A_{1}, (A_{1} ∩ Φ_{6}); R)) is trivial.

The space (A_{1} ∩ Φ_{6})\Λ_{4,2}(l_{1}) also is a fiber bundle with base (K\l_{1}) ∼ C^{1}; its fiber is the space ∂Λ_{4,2}(l_{1}) ∼ (S^{2})^{∗3}. By Lemma 3 the Borel–Moore homology group of this fiber also is trivial. Thus H_{∗}(A_{1}) coincides with the homology group of the space Λ_{4,2}(l_{1}), which is contractible, and acyclicity of A_{1} is completely proved. The proof for A_{2} is exactly the same.

Finally, A_{1} ∩ A_{2} is the continuous order complex of following sets:

a) any point of K;

b) any pair of points in K lying in different lines (one of which can coincide with the intersection point l_{1} ∩ l_{2});

c) any triple of points in K, the first of which is the crossing point l_{1} ∩ l_{2}, and two other lie in different lines l_{1}, l_{2}.

The homology groups of this complex can be easily calculated and coincide with the homology groups of a point. Lemma 8 is completely proved and the column p = 12 actually is empty.

As in the proof of Theorem 2, the ultimate column p = 13 is calculated by a spectral sequence, whose twelve columns repeat these of the main sequence, with coordinates q decreased by twice the number shown in angular parentheses in Proposition 6. The fact that its lower terms kill one another is proved in exactly the same way as it was done in the proof of Theorem 2, see Lemma 5. (The complement of the discriminant is a Stein manifold and cannot have too low-dimensional homology groups).

Finally, the fact that the term E^{1}, shown in Fig. 4, coincides with the stable term E^{∞}, also is proved similarly to the similar statement (Lemma 6) in the proof of Theorem 2. Say, the cell E_{2,23} cannot disappear, because it should match the cell E_{1,26}, which, in its turn, cannot disappear by dimensional reasons. Cell E_{13,9} cannot disappear, because it should match one of cells E_{2,21} or E_{10,13}, etc. □

6 Non-singular cubical surfaces in CP^{3}

Theorem 4. The Poincaré polynomial of the real cohomology group of the space N_{3,3} of nonsingular cubical hypersurfaces in CP^{3} is equal to (1 + t^{3})(1 + t^{5})(1 + t^{7}). In particular, again H^{∗}(N_{3,3}, R) ≃ H^{∗}(P GL(CP^{3}), R).

Proposition 8. There are exactly the following possible singular sets in CP^{3} of homogeneous cubical polynomials in C^{4}:

1. Any point in CP^{3} ⟨16⟩

2. Any pair of points ⟨12⟩

3. Any line ⟨10⟩

4. Any triple of points in general position (i.e. not on the same line) ⟨8⟩

5. Any generic quadric inside any plane CP^{2} ⊂ CP^{3} ⟨5⟩

6. Any pair of intersecting lines in CP^{3} ⟨5⟩

7. Any generic quadruple of points (i.e. not lying in the same two-dimensional plane) ⟨4⟩

8. Any plane CP^{2} ⊂ CP^{3} ⟨4⟩

9. Any triple of lines having one common point (such a polynomial is equal to F = xyz in appropriate coordinates) ⟨1⟩

10. Any generic plane quadric (as in item 5) plus any point not in the same plane ⟨1⟩

11. Entire CP^{3}. ⟨0⟩

Exactly as in the proof of Theorem 3, for any i = 1, …, 11, denote by F_{i} the union of all sets σ_{3,3}(K) ⊂ σ_{3,3} where K is one of sets described in items 1, …, i of the previous proposition.

Proposition 9. The term E^{1} of the spectral sequence, generated by this filtration and converging to the group H̄_{∗}(σ_{3,3}, R), is as shown in Fig. 5. This spectral sequence degenerates at this term: E^{∞} ≡ E^{1}.

Proof. The shape of columns p = 1 and 2 follows immediately from Lemma 2B. The justification of columns p = 4 and 7 needs additionally the following lemma.

Denote by B̃(CP^{3}, k) the subset in B(CP^{3}, k), consisting of generic configurations, i.e. of such that none their three points lie on the same line and none four lie on the same 2-plane.

Lemma 9. There are isomorphisms

H̄_{∗}(B(CP^{3}, 3), ±R) ⟶^{∼} H̄_{∗}(B̃(CP^{3}, 3), ±R),  (13)
H̄_{∗}(B(CP^{3}, 4), ±R) ⟶^{∼} H̄_{∗}(B̃(CP^{3}, 4), ±R),  (14)

induced by identical embeddings. Namely, both groups (13) are isomorphic to R in dimensions 6,8,10 and 12, and are trivial in all other dimensions, and groups (14)

[figure: 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; R; R; R; R; R; R; R; R; R; R; R; R; R^{2}; R; p; q]

Figure 5: Spectral sequence for nonsingular cubical polynomials in C^{4} are trivial in all dimensions other than 12 and are isomorphic to R in dimension 12.

Proof. The isomorphism (13) follows from exactly the same arguments as (9), only with space CP^{2∨} of all lines in CP^{2} replaced by the space G_{2}(C^{4}) of lines in CP^{3}.

The space B(CP^{3}, 4)\B̃(CP^{3}, 4) is the union of two pieces, consisting of 4configurations, whose projective span is a 2-plane (respectively, a line) in CP^{3}. By Lemmas 2 B) and 4 the ±R-homology groups of both these pieces are trivial, and Lemma 9 is proved.

For K of type 3, K = CP^{1} ⊂ CP^{2}, ∂Λ_{3,3}(K) ≃ (S^{2})^{∗2}, therefore the shape of the column p = 3 follows from Lemma 3.

Similarly, for K of type 5 ∂Λ_{3,3}(K) ≃ (S^{2})^{∗3}, and the shape of the column p = 5 follows from the same lemma.

For K of type 6, K = l_{1} ∪ l_{2}, the link ∂Λ_{3,3}(K) is covered by three subsets: A_{1} = Λ_{3,3}(l_{1}), A_{2} = Λ_{3,3}(l_{2}), and A_{3} swept out by all order complexes Λ_{3,3}(κ), where κ are 3-point sets, one whose point is the crossing point l_{1} ∩ l_{2}, and two other lie in different lines l_{1} and l_{2}. The order complexes A_{1} and A_{2} are contractible, and their intersection consists of one point, thus H_{∗}(A_{1} ∪A_{2}) ≃ H_{∗}(pt). Further, A_{3}\(A_{1} ∪A_{2}) is a fiber bundle, whose base is the space C^{1} × C^{1} ≡ (K\l_{1}) × (K\l_{2}) parametrizing all above-described sets κ, and the fiber over such a set is the triangle with two sides removed. Thus the Borel–Moore homology group of such a fiber is trivial, and by the Künneth formula

H̄_{∗}(∂Λ_{3,3}(K)\(A_{1} ∪ A_{2})) ≡ H_{∗}(∂Λ_{3,3}(K), (A_{1} ∪ A_{2})) ≡ 0.

This proves the triviality of the column p = 6.

Similarly, if K is of type 9, K = l_{1} ∩ l_{2} ∩ l_{3}, then the link ∂Λ_{3,3}(K) is covered by four sets A_{1}, A_{2}, A_{3}, A_{4}, where A_{i}, i = 1, 2,and 3, are the order complexes Λ_{3,3}(μ_{i}), μ_{i} are unions of some two of three lines constituting K. A_{4} is swept out by all order complexes Λ_{3,3}(κ), where κ are 4-point sets, one whose point is the crucial point of K, and three other lie in different lines l_{1}, l_{2} and l_{3}. A_{4}\(A_{1} ∪ A_{2} ∪ A_{3}) is a fiber bundle, whose base is the space C^{1} × C^{1} × C^{1} parametrizing all above-described sets κ, and the fiber over such a set is the tetrahedron with three maximal faces removed. Thus the Borel–Moore homology group of such a fiber is trivial, and by the Künneth formula

H̄_{∗}(∂Λ_{3,3}(K)\(A_{1} ∪ A_{2} ∪ A_{3})) ≡ H_{∗}(∂Λ_{3,3}(K), (A_{1} ∪ A_{2} ∪ A_{3})) ≡ 0.

Further, the order complexes A_{1}, A_{2} and A_{3} are acyclic, their pairwise intersections are the order complexes Λ_{3,3}(l_{j}), which also are acyclic, and their total intersection is a point. This proves the triviality of the column p = 9.

For K of type 10, (i.e. consisting of a generic plane quadric K̃ and one point not in the same plane), the link ∂Λ_{3,3}(K) is homeomorphic to the suspension Σ(∂Λ_{3,3}(K̃)), cf. Lemma 7. This proves the triviality of the column p = 10.

Column p = 8. Let K = CP^{2} ⊂ CP^{3}. The link ∂Λ_{3,3}(K) is filtered by its intersections with terms Φ_{i} of the filtration of Λ_{3,3}. We need only the following segment of this filtration:

∂Λ_{3,3}(K) ∩ Φ_{4} ⊂ ∂Λ_{3,3}(K) ∩ Φ_{5} ⊂ ∂Λ_{3,3}(K) ∩ Φ_{6} ≡ ∂Λ_{3,3}(K).  (15)

The first term of this filtration, ∂Λ_{3,3}(K) ∩ Φ_{4}, is acyclic by Lemma 5. The set (∂Λ_{3,3}(K) ∩ Φ_{5})\(∂Λ_{3,3}(K) ∩ Φ_{4}) is the space of a fiber bundle, whose base is the space of all generic quadrics in CP^{2} (studied in § 2), and the fiber over such a quadric κ is the open cone Λ^{∘}_{3,3} (κ). The Borel–Moore homology group of this fiber coincides (up to a shift of dimensions) with the (reduced modulo a point) homology group of the corresponding link ∂Λ_{3,3}(κ) ∼ (S^{2})^{∗3}, which is trivial by Lemma 3. Finally, the set (∂Λ_{3,3}(K) ∩ Φ_{6})\(∂Λ_{3,3}(K) ∩ Φ_{5}) is the space of a fiber bundle, whose base is the space B(CP^{2∨}, 2) of pairs of complex lines in CP^{2}, and the fiber over such a pair κ is the open cone Λ^{∘}_{3,3} (κ). By the boundary isomorphism (5), to find the Borel–Moore homology of such a fiber, we need only to calculate the (usual) homology of the link ∂Λ_{3,3}(κ), which already was calculated (see the study of the column p = 6) and is trivial.

This proves the triviality of the column p = 8 and terminates the justification of columns 1 through 10 of Fig. 5. The triviality of the column p = 11 and the degeneration of the spectral sequence at the term E^{1} follow from exactly the same reasons as in the proofs of previous theorems: the former follows from the fact that Π_{3,3}\Σ_{3,3} is a Stein manifold, and the latter from the fact that it is divisible by C^{∗} (and from the explicit shape of the term E^{1}).

Proposition 9 and Theorem 4 are completely proved.

7 Non-degenerate quadratic vector fields in C^{3}

Definition. A collection of three homogeneous functions (v_{1}, v_{2}, v_{3}) : C^{3} → C^{3} is nondegenerate, if they vanish simultaneously only at the origin of the issue space.

Theorem 5. The Poincaré polynomial of the rational cohomology group of the space of nondegenerate triples of homogeneous functions of degrees 2 in C^{3} is equal to (1 + t)(1 + t^{3})(1 + t^{5}).

Proof. Denote by Σ(2, 2, 2) the discriminant subset in the space C^{18} of all triples of quadratic functions in C^{3}, i.e. the set of all degenerate triples. The conical resolution of this subset is constructed in exactly the same way as similar resolutions of spaces Σ_{d,n} described in § 3; instead of possible singular subsets in CP^{n} we use possible sets of common zeros of a triple of polynomials.

Proposition 10. There are exactly the following possible sets in CP^{2} of common zeros of triples of homogeneous polynomials of degree 2 in C^{3}:

1. Any point in CP^{2} ⟨15⟩

[figure: p; q; R; R; R; R; R; R; R; 5; 4; 3; 2; 1; 33; 32; 31; 30; 29; 28; 27; 26; 25; 24; 23; 6; 7; 8; 9]

Figure 6: Spectral sequence for nondegenerate quadratic vector fields in C^{3}

2. Any pair of points ⟨12⟩

3. Any generic triple of points (i.e. not lying on the same line) ⟨9⟩

4. Any line CP^{1} ⊂ CP^{2} ⟨9⟩

5. Any generic quadruple of points ⟨6⟩

6. Any line CP^{1} ⊂ CP^{2} plus a point outside it ⟨6⟩

7. Any nonsingular quadric in CP^{2} ⟨3⟩

8. Two lines in CP^{2} ⟨3⟩

9. Entire CP^{2}. ⟨0⟩ □

Proposition 11. The term E^{1} of the spectral sequence, generated by the corresponding filtration and converging to the group H̄_{∗}(Σ_{(}2, 2, 2)R), is as shown in Fig. 6. This spectral sequence degenerates at this term: E^{∞} ≡ E^{1}.

The proof of this proposition repeats essentially some fragments of the proof of Proposition 7 and will be omitted. The comparison of spectral sequences of Figs. 2 and 6 implies the following version of the (homological) Smale–Hirsch principle.

Proposition 12. The gradient embedding C^{10} → C^{18}, mapping any homogeneous polynomial C^{3} → C^{1} of degree 3 to the collection of its three partial derivatives, induces the isomorphism between rational cohomology groups of spaces of nonsingular cubical polynomials and nondegenerate quadratic vector fields in C^{3}. □

8 Problems and concluding remarks

1. To explain the 6-dimensional generator of H^{∗}(N_{4,2}).

2. To find general properties of the above calculations. Of course, these calculations (and the number of strata to be considered) grow at least exponentially together with n and d. Note however that all the infinite strata have contributed nothing in our spectral sequences. Is it so also for greater n and d?

Is H^{∗}(N_{3,n}, R) ≃ H^{∗}(P GL(CP^{n}), R) for any n?

3. To make all the above calculations for the cohomology groups with coefficients in Z or at least Z_{2}. The same for the real version of the problem, i.e. for spectral sequences calculating the cohomology of spaces of nonsingular algebraic hypersurfaces in RP^{n}.

The involution of complex conjugation acts naturally on our spectral sequences, thus they are a good tool to check the M-property for such spaces or some strata of the discriminant.

However, the most useful for the original problem of the rigid homotopy classification should be the similar calculation of R-valued cohomology groups of spaces of real nonsingular objects. Note that V. Kharlamov [7] considered the topology of discriminant sets in a connection with the rigid classification problem.

Problem: to express the standard invariants of rigid isotopy classification of projective curves and surfaces in terms of dual homology classes of discriminants.

4. A collection of points in RP^{2} or in CP^{2} is called d-sufficient (respectively, d-degenerate), if any homogeneous polynomial of degree d with singular points in it vanishes (respectively, has singular points) at entire curve of smaller degree passing through some of them. The simplest example: any [d/2] + 1 (respectively, d) points, lying on the same line, form a d-sufficient (respectively, d-degenerate) collection.

Problem. To study the set of such configurations. For which smallest d there are minimal d-degenerate configurations such that the corresponding curve of singularities contains not all their points? For which i the spaces of all i-point dnondegenerate configurations are disconnected (both in real and complex versions) and how many components do they have?

To describe the class of algebraic subsets in RP^{2} or CP^{2} which satisfy the following d-overdeterminacy property: any homogeneous polynomial of degree d, having singular points at this subset, is identically equal to 0.

More generally, let us call a point y ∈ P^{2} a d-consequence of points x_{1}, …, x_{l}, if any homogeneous polynomial of degree d, having singularities at these l points, has also a singularity at the point y. E.g., if (x_{1}, …, x_{l}) is a d-degenerate (respectively, d-overdetermined) set, then all points of the corresponding curve (respectively, all points at all) are its d-consequences. Are there other examples?

Consider some natural family of algebraic subsets in P^{2}, e.g. the set of all collections, consisting of several curves of fixed degrees and several points not on these curves. Any point of this family defines a subspace in Π_{d}: the space of polynomials having singularities at all points of the corresponding algebraic set. Is it possible to describe easily the set of points of our family, for which the dimension of this subspace “jumps”?

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