Homology of Spaces of Non-Resultant Homogeneous Polynomial Systems in ${\mathbb R}^2$ and ${\mathbb C}^2$

Abstract

The resultant variety in the space of systems of homogeneous polynomials of some given degrees consists of such systems having non-trivial solutions. We calculate the integer cohomology groups of all spaces of non-resultant systems of polynomials ${\mathbb R}^2 \rightarrow {\mathbb R}$, and also the rational cohomology rings of spaces of non-resultant systems and non-m-discriminant polynomials in ${\mathbb C}^2$.

1 Introduction

Given n natural numbers $d_1 \ge d_2 \ge \dots \ge d_n$, consider the space of all real homogeneous polynomial systems

$$\begin{aligned} \left\{ \begin{aligned} a_{1,0} x^{d_1} + a_{1,1} x^{d_1-1}y + \dots + a_{1,{d_1}}y^{d_1} \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ a_{n,0} x^{d_n} + a_{n,1} x^{d_n-1}y + \dots + a_{n,{d_n}}y^{d_n} \\ \end{aligned} \right. \end{aligned}$$

(1)

in two real variables x, y.

We will refer to this space as ${\mathbb R}^D$, $D = \sum _1^n (d_i+1)$. The resultant variety $\Sigma \subset {\mathbb R}^D$ is the space of all systems having non-zero solutions. $\Sigma$ is a semialgebraic subvariety of codimension $n-1$ in ${\mathbb R}^D$.

Below we calculate the cohomology group of its complement, $H^*({\mathbb R}^D {\setminus } \Sigma )$. Also, we calculate the rational cohomology rings of the complex analogs ${\mathbb C}^D {\setminus } \Sigma _{\mathbb C}$ of all spaces ${\mathbb R}^D {\setminus } \Sigma$.

For the “affine” version of the “real” problem (concerning the space of non-resultant systems of polynomials ${\mathbb R}^1 \rightarrow {\mathbb R}^1$ with leading terms $x^{d_i}$), see, e.g., Vassiliev (1994, 1997) and Kozlowski and Yamaguchi (2000); for the “complex” problem with $n = 2$ see also Cohen et al. (1991). A similar calculation for spaces of real homogeneous polynomials in ${\mathbb R}^2$ without zeros of multiplicity $\ge m$ was done in Vassiliev (1998).

The entire study of homology groups of spaces of non-singular (in appropriate sense) objects goes back to the Arnold’s works (1970, 1989), as well as the idea of using the Alexander duality in this problem.

2 Main Results

2.1 Notation

For any natural p, denote by N(p) the sum of all numbers $d_i+1,$ $i=1, \dots , n,$ which are less than or equal to p, plus p times the number of those $d_i$ which are equal to or greater than p. [In other words, N(p) is the area of the part of Young diagram $(d_1+1, \dots , d_n+1)$ strictly to the left from the $(p+1)$-th column.] Let the index $\Upsilon (p)$ be equal to the number of even numbers $d_i \ge p$ if p is even, and to the number of odd numbers $d_i \ge p$ if p is odd. By $\tilde{H}^*(X)$ we denote the cohomology group reduced modulo a point. ${\overline{H}}_*(X)$ denotes the Borel–Moore homology group, i.e. the homology group of the complex of locally finite singular chains of X.

Theorem 1

If the space ${\mathbb R}^D {\setminus } \Sigma$ is non-empty (i.e. either $n>1$ or $d_1$ is even), then the group $\tilde{H}^*({\mathbb R}^D {\setminus } \Sigma , {\mathbb Z})$ is equal to the direct sum of following groups:

A) For any $p=1, \dots , d_3$,

if $\Upsilon (p)$ is even, then ${\mathbb Z}$ in dimension $N(p)-2p$ and ${\mathbb Z}$ in dimension $N(p)-2p+1$,

if $\Upsilon (p)$ is odd, then only one group ${\mathbb Z}_2$ in dimension $N(p)-2p+1$;

B) If $d_1-d_2$ is odd, then an additional summand ${\mathbb Z}$ in dimension $D-d_1-d_2-2$. If $d_1-d_2$ is even, then an additional summand ${\mathbb Z}^{d_2-d_3+1}$ in dimension $D-d_1-d_2-1$ and (if $d_2 \ne d_3)$ a summand ${\mathbb Z}^{d_2-d_3}$ in dimension $D-d_1-d_2-2$.

Example 1

Let $n=2$ [so that part (A) in the statement of Theorem 1 is void]. If $d_1$ and $d_2$ are of the same parity, then ${\mathbb R}^D {\setminus } \Sigma$ consists of $d_2+1$ connected components, each of which is homotopy equivalent to a circle. For an invariant, which separates systems belonging to different components, we can take the index of the induced map of the unit circle $S^1 \subset {\mathbb R}^2$ into ${\mathbb R}^2 {\setminus } 0$. This index can take all values of the same parity as $d_1$ and $d_2$ from the segment $[-d_2,d_2]$. The 1-dimensional cohomology class inside any component is just the rotation number of the image of a fixed point [say, (1, 0)] around the origin. Moreover, the images of this point under our non-resultant systems define a map ${\mathbb R}^D {\setminus } \Sigma \rightarrow {\mathbb R}^2 {\setminus } 0;$ it is easy to see that any fiber of this map consists of $d_2+1$ contractible components.

If $d_1$ and $d_2$ are of different parities, then the space ${\mathbb R}^D {\setminus } \Sigma$ has the homology of a two-point set. The invariant separating its two connected components can be calculated as the parity of the number of zeros of the odd-degree polynomial of our non-resultant system, which lie in the (well-defined) domain in ${\mathbb {RP}}^1$ where the even-degree polynomial is positive.

Now, let ${\mathbb C}^D$ be the space of all polynomial systems (1) with complex coefficients $a_{i,j}$, and $\Sigma _{\mathbb C}\subset {\mathbb C}^D$ the set of systems having solutions in ${\mathbb C}^2 {\setminus } 0$.

Theorem 2

For any $n>1$, the ring $H^*({\mathbb C}^D {\setminus } \Sigma _{{\mathbb C}}, {\mathbb Q})$ is an exterior algebra over ${\mathbb Q}$ with two generators of dimensions $2n-3$ and $2n-1$. Namely, these generators are the linking number with the Borel–Moore fundamental class of entire resultant variety and the pull-back of the basic cohomology class under the map ${\mathbb C}^D {\setminus } \Sigma _{\mathbb C}\rightarrow {\mathbb C}^n {\setminus } 0$ defined by restrictions of non-resultant systems $(f_1, \dots , f_n)$ to the point (1, 0). The weight filtrations of these two generators and their product in the mixed Hodge structure of ${\mathbb C}^D {\setminus } \Sigma _{\mathbb C}$ are equal to $2n-2$, 2n and $4n-2$ respectively.

Consider also the space ${\mathbb C}^{d+1}$ of all complex homogeneous polynomials

$$\begin{aligned} a_0 x^{d} + a_1 x^{d-1}y + \dots + a_dy^{d}{,} \end{aligned}$$

and m-discriminant $\Sigma _m$ in it consisting of all polynomials vanishing on some line with multiplicity $\ge m$.

Theorem 3

For any $m>1$ and $d \ge 2m-1$, the ring $H^*({\mathbb C}^{d+1} {\setminus } \Sigma _m, {\mathbb Q})$ is isomorphic to an exterior algebra over ${\mathbb Q}$ with two generators of dimensions $2m-3$ and $2m-1$. The weight filtrations of these two generators and of their product are equal to $2m-2$, 2m and $4m-2$ respectively. For any $m>1$ and $d \in [m+1, 2m-2]$, this ring is isomorphic to ${\mathbb Q}$ in dimensions $0, 2m-3, 2m-1$ and $2d-2$, and is trivial in all other dimensions; the multiplication is obviously trivial. For $d=m>1$ this ring is isomorphic to ${\mathbb Q}$ in dimensions 0 and $2m-3$, and is trivial in all other dimensions.

3 Some Preliminary Facts

Denote by B(M, p) the configuration space of subsets of cardinality p of a topological space M.

Lemma 1

For any natural p,  there is a locally trivial fiber bundle $B(S^1,p) \rightarrow S^1$ whose fiber is homeomorphic to ${\mathbb R}^{p-1}$. This fiber bundle is non-orientable if p is even, and is orientable (and hence trivial) if p is odd.

Indeed, the projection of this fiber bundle can be realised as the product of p points of the unit circle in ${\mathbb C}^1$. The fiber of this bundle can be identified in terms of the universal covering ${\mathbb R}^p \rightarrow T^p$ with any connected component of some hyperplane $\{x_1 + \dots + x_p = \text{ const }\}$, from which all affine planes given by $x_i = x_j + 2\pi k$, $i \ne j$, $k \in {\mathbb Z}$, are removed. Such a component is convex and hence diffeomorphic to ${\mathbb R}^{p-1}$. The assertion on orientability can be checked immediately. $\square$

Let us embed a manifold M generically into the space ${\mathbb R}^T$ of a very large dimension, and denote by $M^{*r}$ the union of all $(r-1)$-dimensional simplices in ${\mathbb R}^T$, whose vertices lie in this embedded manifold (and the “genericity” of the embedding means that if two such simplices have a common point in ${\mathbb R}^T$, then their minimal faces containing this point coincide).

Proposition 1

(C. Caratheodory theorem: see also Vassiliev 1997; Kallel and Karoui 2011) For any $r \ge 1$, the space $(S^1)^{*r}$ is homeomorphic to $S^{2r-1}$.

Remark 1

This homeomorphism can be realized as follows. Consider the space ${\mathbb R}^{2r+1}$ of all real homogeneous polynomials ${\mathbb R}^2 \rightarrow {\mathbb R}^1$ of degree 2r, the convex cone in this space consisting of everywhere non-negative polynomials, and (also convex) dual cone in the dual space $\widehat{\mathbb R}^{2r+1}$ consisting of linear forms taking only positive values inside the previous cone. The intersection of the boundary of this dual cone with the unit sphere in $\widehat{\mathbb R}^{2r+1}$ is naturally homeomorphic to $(S^1)^{*r}$; on the other hand it is homeomorphic to the boundary of a convex 2r-dimensional domain.

Lemma 2

(see Vassiliev 1999, Lemma 3) For any $r>1$, the group $H_*((S^2)^{*r}, {\mathbb Q})$ is trivial in all positive dimensions. $\square$

Consider the “sign local system” $\pm {\mathbb Q}$ over $B({\mathbb {CP}}^1,p)$, i.e. the local system of groups with fiber ${\mathbb Q}$ such that the elements of $\pi _1(B({\mathbb {CP}}^1,p))$ defining odd (respectively, even) permutations of p points in ${\mathbb {CP}}^1$ act in the fiber as multiplication by $-1$ (respectively, by 1).

Lemma 3

(see Vassiliev 1999, Lemma 2) All Borel–Moore homology groups $\overline{H}_i(B({\mathbb {CP}}^1,p);\pm {\mathbb Q})$ with $p \ge 1$ are trivial except

$$\begin{aligned} \overline{H}_0(B({\mathbb {CP}}^1,1), \pm {\mathbb Q}) \,{\cong }\, \overline{H}_2(B({\mathbb {CP}}^1,1), \pm {\mathbb Q}) \,{\cong }\, \overline{H}_2(B({\mathbb {CP}}^1,2), \pm {\mathbb Q}) \,{\cong }\, {\mathbb Q}. \end{aligned}$$

$\square$

4 Proof of Theorem 1

Following Arnold (1970), we use the Alexander duality

$$\begin{aligned} \tilde{H}^i({\mathbb R}^D {\setminus }\Sigma ) \simeq {\overline{H}} _{D-i-1}(\Sigma ). \end{aligned}$$

(2)

4.1 Simplicial Resolution of $\Sigma$

To calculate the right-hand group in (2), we construct a resolution of the space $\Sigma$. Let $\chi : {\mathbb {RP}}^1 \rightarrow {\mathbb R}^T$ be a generic embedding, $T\gg d_1$. For any system $\Phi =(f_1, \dots , f_n) \in \Sigma$ not equal identically to zero, consider the simplex $\Delta (\Phi )$ in ${\mathbb R}^T$ spanned by the images $\chi (x_i)$ of all points $x_i \in {\mathbb {RP}}^1$ corresponding to all lines, on which the system f has a common root. (The maximal possible number of such lines is obviously equal to $d_1.$)

Furthermore, consider a subset in the direct product ${\mathbb R}^D \times {\mathbb R}^T$, namely, the union of all simplices of the form $\Phi \times \Delta (\Phi ),$ $\Phi \in \Sigma {\setminus }0$. This union is not closed: the set of its limit points not belonging to it is the product of the point $0 \in {\mathbb R}^D$ (corresponding to the zero system) and the union of all simplices in ${\mathbb R}^T$ spanned by the images of no more than $d_1$ different points of the line ${\mathbb {RP}}^1.$ By the Caratheodory theorem, the latter union is homeomorphic to the sphere $S^{2d_1-1}.$ We can assume that our embedding $\chi : {\mathbb {RP}}^1 \rightarrow {\mathbb R}^T$ is algebraic, and hence this sphere is semialgebraic. Take a generic $2d_1$-dimensional semialgebraic disc in ${\mathbb R}^T$ bounded by this sphere (e.g., the union of segments connecting the points of this sphere with a generic point in ${\mathbb R}^T$), and add the product of the point $0 \in {\mathbb R}^D$ and this disc to the previous union of simplices $\Phi \times \Delta (\Phi ) \subset {\mathbb R}^D \times {\mathbb R}^T$. The resulting closed subset in ${\mathbb R}^D \times {\mathbb R}^T$ will be denoted by $\sigma$ and called a simplicial resolution of $\Sigma$.

Lemma 4

The obvious projection $\sigma \rightarrow \Sigma$ (induced by the projection of ${\mathbb R}^D \times {\mathbb R}^T$ onto the first factor) is proper, and the induced map between one-point compactifications of these spaces is a homotopy equivalence.

This follows easily from the fact that this projection is a stratified map of semialgebraic spaces, and the preimage of any point of $\Sigma$ is contractible: see Vassiliev (1994, 1997). $\square$

So, we can (and will) calculate the group ${\overline{H}} _*(\sigma )$ instead of ${\overline{H}} _*(\Sigma )$.

Remark 2

There is a different construction of a simplicial resolution of $\Sigma$ in terms of “Hilbert schemes”. Namely, let $I_p$ be the space of all ideals of codimension p in the space of smooth functions ${\mathbb {RP}}^1 \rightarrow {\mathbb R}^1$ equipped with the natural “Grassmannian” topology. It is easy to see that $I_p$ is homeomorphic to the p-th symmetric power $S^p({\mathbb {RP}}^1) = ({\mathbb {RP}}^1)^p/S(p){;}$ in particular, it contains the configuration space $B({\mathbb {RP}}^1,p)$ as an open dense subset. Consider the disjoint union of these $d_1$ spaces $I_1, \dots , I_{d_1}$ augmented with the one-point set $I_\infty$ symbolizing the zero ideal. The incidence of ideals makes this union a partially ordered set. Consider the continuous order complex $\Xi _{d_1}$ of this poset, i.e. the subset in the join $I_1 * \dots * I_{d_1} * I_\infty$ consisting of simplices, whose all vertices are incident to one another. For any polynomial system $\Phi =(f_1, \dots , f_n) \in {\mathbb R}^D$, denote by $\Xi (\Phi )$ the subcomplex in $\Xi _{d_1}$ consisting of all simplices, whose all vertices correspond to ideals containing all polynomials $f_1, \dots , f_n$. The simplicial resolution $\tilde{\sigma }\subset \Sigma \times \Xi _{d_1}$ is defined as the union of simplices $\Phi \times \Xi (\Phi )$ over all $\Phi \in \Sigma$.

This construction is homotopy equivalent to the previous one. In particular, the Caratheodory theorem has the following version (see Kallel and Karoui (2011)): the continuous order complex of the poset of all ideals of codimension $\le r$ in the space of functions $S^1 \rightarrow {\mathbb R}^1$ is homotopy equivalent to $S^{2r-1}$.

However, this construction is less convenient for our practical calculations than the one described above and used previously in Vassiliev (1994, 1999) [and extended to some more complicated situations in Gorinov (2005)].

The space $\sigma$ has a natural increasing filtration $F_1 \subset \dots \subset F_{d_1+1} = \sigma$: its term $F_p,$ $p \le d_1,$ is the closure of the union of all simplices of the form $\Phi \times \Delta (\Phi )$ over all polynomial systems $\Phi$ having no more than p lines of common zeros. Alternatively, it can be described as the union of all no more than $(p-1)$-dimensional faces of all simplices $\Phi \times \Delta (\Phi )$ over all systems $\Phi \in \Sigma {\setminus } 0$, completed with all no more than $(p-1)$-dimensional simplices spanning some $\le p$ points of the manifold $\{0\} \times \chi (\mathbb {RP}^1)$.

Lemma 5

For any $p =1, \ldots , d_1,$ the term $F_p {\setminus }F_{p-1}$ of our filtration is the space of a locally trivial fiber bundle over the configuration space $B({\mathbb {RP}}^1,p),$ with fibers equal to the direct product of a $(p-1)$-dimensional open simplex and a $(D-N(p))$-dimensional real space. The corresponding bundle of open simplices is orientable if and only if p is odd (i.e. exactly when the base configuration space is orientable), and the bundle of $(D-N(p))$-dimensional spaces is orientable if and only if the index $\Upsilon (p)$ is even.

The last term $F_{d_1+1} {\setminus }F_{d_1}$ of this filtration is homeomorphic to an open $2d_1$-dimensional disc.

Indeed, to any configuration $(x_1, \ldots , x_p) \in B({\mathbb {RP}}^1,p),$ $p \le d_1$, there corresponds the direct product of the interior part of the simplex in ${\mathbb R}^T$ spanned by the images $\chi (x_i)$ of points of this configuration, and the subspace of ${\mathbb R}^D$ consisting of polynomial systems that have solutions on corresponding p lines in ${\mathbb R}^2.$ The codimension of the latter subspace is equal exactly to N(p). The assertion concerning the orientations can be checked in a straightforward way. The description of $F_{d_1+1} {\setminus }F_{d_1}$ follows immediately from the construction and the Caratheodory theorem. $\square$

Consider the spectral sequence $E_{p,q}^r,$ calculating the group ${\overline{H}} _*(\Sigma )$ and generated by this filtration. Its term $E_{p,q}^1$ is canonically isomorphic to the group ${\overline{H}} _{p+q}(F_p {\setminus }F_{p-1}).$ By Lemma 5, its column $E_{p,*}^1,$ $p \le d_1,$ is as follows. If $\Upsilon (p)$ is even, then this column contains exactly two non-trivial terms $E_{p,q}^1$, both isomorphic to ${\mathbb Z}$, for q equal to $D-N(p)+p-1$ and $D-N(p)+p-2$. If $\Upsilon (p)$ is odd, then this column contains only one non-trivial term $E_{p,q}^1$ isomorphic to ${\mathbb Z}_2$, for $q=D-N(p)+p-2$. Finally, the column $E^1_{d_1+1,*}$ contains only one non-trivial element $E^1_{d_1+1,d_1-1} \,{\cong }\, {\mathbb Z}$.

Before calculating the differentials and further terms $E^r$, $r >1$, let us consider several basic examples.

4.2 The Case $n=1$

If our system consists of only one polynomial of degree $d_1$, then the term $E^1$ of our spectral sequence looks as in Fig. 1; in particular, all non-trivial groups $E_{p,q}^1$ lie in two rows $q=d_1$ and $q=d_1-1$.

Fig. 1

$E^1$ for $n=1$, $d_1$ even (left) and $n=1$, $d_1$ odd (right)

Lemma 6

If $n=1$, then in both cases of even or odd $d_1$, all possible horizontal differentials $\partial _1: E_{p,d_1-1}^1 \rightarrow E_{p-1,d_1-1}^1$ of the form ${\mathbb Z}\rightarrow {\mathbb Z}_2$, $p=d_1+1, d_1-1, d_1-3, \dots$ are epimorphisms, and all differentials $\partial _2: E_{p,d_1-1}^2 \rightarrow E_{p-2,d_1}^2$ of the form ${\mathbb Z}\rightarrow {\mathbb Z}$, $p=d_1+1, d_1-1, d_1-3, \dots$ are isomorphisms. In particular, the unique surviving term $E_{p,q}^3$ for the “even” spectral sequence is $E_{1,d_1-1}^3\,{\cong }\, {\mathbb Z}$, and for the “odd” one it is $E_{2,d_1-1}^3 \,{\cong }\, {\mathbb Z}$.

Indeed, in both cases we know the answer. In the “odd” case, the discriminant coincides with entire ${\mathbb R}^D = {\mathbb R}^{d_1+1}$. In the “even” case, its complement consists of two contractible components, so that ${\overline{H}} _*(\Sigma ) = {\mathbb Z}$ in dimension $d_1$ and is trivial in all other dimensions. Therefore, all terms $E_{p,q}$ with $p+q$ not equal to $d_1+1$ (respectively, to $d_1$) in the odd- (respectively, even-) dimensional case should die at some stage; this is possible only if all assertions of our lemma hold. $\square$

4.3 The Case $n=2$

There are two very different situations depending on the parity of $d_1-d_2$. In Fig. 2, we demonstrate these situations in two particular cases: $(d_1,d_2)=(6,3)$ and (7, 3). However, the general situation is essentially the same; namely, the following is true.

Fig. 2

$E^1$ for $n=2$, $d_1-d_2$ odd (left) and $n=2$, $d_1-d_2$ even (right)

If $n=2$ and $d_1-d_2$ is odd, then all indices $\Upsilon (p)$, $p=1, \dots , d_2+1$, are odd, and hence all non-trivial groups $E_{p,q}^1$ with such p lie on the line $\{p+q=d_1+d_2\}$ only and are equal to ${\mathbb Z}_2$.

If $n=2$ and $d_1-d_2$ is even, then all indices $\Upsilon (p)$, $p=1, \dots , d_2+1$, are even, and hence all non-trivial groups $E_{p,q}^1$ with such p lie on two lines $\{p+q=d_1+d_2\}$, $\{p+q=d_1+d_2+1\}$, and are all equal to ${\mathbb Z}$.

In both cases, all groups $E_{p,q}^1$ with $p > d_2$ are the same as in the case $n=1$ with the same $d_1$. Moreover, the differentials $\partial _1$ and $\partial _2$ between these groups are also the same as for $n=1$; therefore, all of these groups die at $E^3$ except for $E_{d_2+1,d_1-1}^3 \,{\cong }\, {\mathbb Z}$ for even $d_1-d_2$, and $E_{d_2+2,d_1-1}^3\,{\cong }\, {\mathbb Z}$ for odd $d_1-d_2$.

In the case of even $d_1-d_2$, all other differentials between the groups $E_{p,q}^r$ are trivial, because otherwise the group $\tilde{H}^0({\mathbb R}^D {\setminus } \Sigma )$ would be smaller than ${\mathbb Z}^{d_2}$, in contradiction to $d_2+1$ different components of this space indicated in Example 1.

On the contrary, if $d_1-d_2$ is odd, then all the differentials $d_r:E_{d_2+2,d_1-1}^r \rightarrow E_{d_2+2-r, d_1-2+r}^r$, $r=1, \dots , d_1-d_2+1$, are epimorphic just because the integer cohomology group of the topological space ${\mathbb R}^D {\setminus } \Sigma$ cannot have non-trivial torsion subgroup in dimension 1. Therefore, the unique nontrivial group $E_{p,q}^\infty$ in this case is $E_{d_2+2,d_1-1}^\infty \,{\cong }\, {\mathbb Z}$.

This proves Theorem 1 for $n=2$.

4.4 The General Case

Now suppose that our systems (1) consist of $n\ge 3$ polynomials. Let again $\sigma$ be the simplicial resolution of the corresponding resultant variety constructed in Sect. 4.1, and $\sigma '$ be the simplicial resolution of the resultant variety for $n=2$ and the same $d_1$ and $d_2$. The parts $\sigma {\setminus } F_{d_3}(\sigma )$ and $\sigma ' {\setminus } F_{d_3}(\sigma ')$ of these resolutions are canonically homeomorphic to one another as filtered spaces. In particular, $E_{p,q}^1(\sigma ) = E_{p,q}^1(\sigma ')$ if $p > d_3$, and $E_{p,q}^r(\sigma ) = E_{p,q}^r$ if $p \ge d_3+r$. All non-trivial terms $E_{p,q}^r(\sigma )$ with $p \le d_3$ are placed in such a way that no non-trivial differentials $\partial _r$ can act between these terms, as well as no differentials can act to these terms from the cells $E_{p,q}^r$ with $p > d_3$, which have survived the differentials between these cells described in the previous subsection.

Therefore, the final term $E_{p,q}^\infty (\sigma )$ coincides with $E_{p,q}^1(\sigma )$ in the domain $\{p \le d_3\}$, and coincides with the term $E_{p,q}^\infty (\sigma ')$ of the truncated spectral sequence calculating the Borel–Moore homology of $\sigma ' {\setminus } F_{d_3}(\sigma ')$ in the domain $\{p > d_3\}$. This completes the proof of Theorem 1. $\square$

5 Proof of Theorem 2

The simplicial resolution $\sigma _{{\mathbb C}}$ of $\Sigma _{{\mathbb C}}$ appears in the same way as its real analog $\sigma$ in the previous section. It also has a natural filtration $F_1 \subset \dots \subset F_{d_1+1} = \sigma _{{\mathbb C}}$. For $p \in [1, d_1]$, its term $F_p {\setminus } F_{p-1}$ is fibered over the configuration space $B({\mathbb {CP}}^1,p)$; its fiber over a configuration $(x_1, \dots x_p)$ is equal to the product of the space ${\mathbb C}^{D-N(p)}$ (consisting of all complex systems (1) vanishing at all lines corresponding to the points of this configuration) and the $(p-1)$-dimensional simplex whose vertices correspond to the points of the configuration. In particular, our spectral sequence calculating rational Borel-Moore homology of $\sigma _{{\mathbb C}}$ has $E^1_{p,q} \,{\cong }\, {\overline{H}} _{q-2(D-N(p))+1}(B({\mathbb {CP}}^1,p); \pm {\mathbb Q})$ for such p. By Lemma 3, only the following such groups are non-trivial: $E^1_{1,2(D-n)-1} \,{\cong }\, {\mathbb Q}$, $E^1_{1,2(D-n)+1} \,{\cong }\, {\mathbb Q}$, and (if $d_1 >1$) $E^1_{2, 2(D-2n)+1} \,{\cong }\, {\mathbb Q}$.

The last term $F_{d_1+1} {\setminus } F_{d_1}$ is homeomorphic to the cone over the $d_1$-th self-join $({\mathbb {CP}}^1)^{*d_1}$ with the base of this cone removed (as it belongs to $F_{d_1}$). Therefore, by Lemma 2, the column $E^1_{d_1+1,*}$ is trivial if $d_1>1$, and contains a unique non-trivial group $E^1_{2,1} \,{\cong }\, {\mathbb Q}$ if $d_1=1$.

So, in any case, the first sheet $E^1$ of our spectral sequence has only three non-trivial terms $E^1_{1,2(D-n)-1}$, $E^1_{1,2(D-n)+1}$, and $E^1_{2, 2(D-2n)+1}$, all of which are isomorphic to ${\mathbb Q}$. The differentials in it are obviously trivial; therefore, the group ${\overline{H}} _*(\sigma )$ has three non-trivial terms in dimensions $2(D-n)$, $2(D-n)+2$, and $2(D-2n)+3$. By Alexander duality in the space ${\mathbb C}^D$, this gives us three groups $\tilde{H}^{2n-3} \,{\cong }\, {\mathbb Q}$, $\tilde{H}^{2n-1} \,{\cong }\, {\mathbb Q}$, and $\tilde{H}^{4n-4}\,{\cong }\, {\mathbb Q}$, and zero in all other dimensions.

All assertions of Theorem 2 concerning the ring structure, realization of cohomology classes, and the weight filtration are well-known or obvious in the case $d_1=1$ (when $D = 2n$ and ${\mathbb C}^{2n} {\setminus } \Sigma _{\mathbb C}$ is the space of pairs of linearly independent vectors in ${\mathbb C}^n$, and is homotopy equivalent to the Stiefel manifold $V_2({\mathbb C}^n)$). The general case can be deduced from this one by the map $P: {\mathbb C}^{2n} {\setminus } \Sigma _{\mathbb C}\rightarrow {\mathbb C}^D {\setminus } \Sigma _{\mathbb C}$ sending any collection of linear functions $(f_1, \dots , f_n)$ to $(f_1^{d_1}, \dots , f_n^{d_n})$. Indeed, the realization of $(2n-1)$-dimensional classes follows from the commutative diagram

$$\begin{aligned} \begin{array}{lcl} {\mathbb C}^{2n} {\setminus } \Sigma _{\mathbb C}&{} \mathop {\longrightarrow }\limits ^{P}&{} {\mathbb C}^D {\setminus } \Sigma _{\mathbb C}\\ \ \downarrow &{} &{} \ \downarrow \\ {\mathbb C}^n {\setminus } 0 &{} \longrightarrow &{} {\mathbb C}^n {\setminus } 0 \end{array}, \end{aligned}$$

where the lower horizontal arrow is defined by

$$\begin{aligned} (z_1, \dots , z_n) \mapsto (z_1^{d_1}, \dots , z_n^{d_n}) \end{aligned}$$

and induces an isomorphism of $(2n-1)$-dimensional rational homology groups. The assertion on the realization of $(2n-3)$-dimensional classes is obvious. The statements on the multiplication and the weight filtration follow from the naturality of these structures. $\Box$

6 Proof of Theorem 3

The additive part of this theorem can be proved in almost the same way as that of Theorem 1: see Vassiliev (1998). In particular, we construct a simplicial resolution $\sigma _m$ of the m-discriminant variety $\Sigma _m$. It has a natural filtration $\Phi _1 \subset \dots \subset \Phi _{[d/m]} \subset \Phi _{[d/m]+1} = \sigma _m$. The term $\Phi _p {\setminus } \Phi _{p-1}$, $p \le [d/m]$, of this filtration is the space of a fiber bundle with the base $B(\mathbb {CP}^1,p)$. Its fiber over the collection of points $(z_1, \dots , z_p) \subset \mathbb {CP}^1$ is the product of an open $(p-1)$-dimensional simplex whose vertices are related with these p points, and the subspace of codimension mp in ${\mathbb C}^{d+1}$ consisting of all polynomials having m-fold zeros on the corresponding p lines. The term $\Phi _{[d/m]+1} {\setminus } \Phi _{[d/m]}$ appears from the zero polynomial and is the cone over the space $(\mathbb {CP}^1)^{*[d/m]}$ with the base of this cone removed. The term $E^1$ of the corresponding spectral sequence can be calculated immediately with the help of Lemmas 2 and 3. Its shape implies that all further differentials of the spectral sequence are trivial, with unique exception in the case $d=m$, when all non-zero (isomorphic to ${\mathbb Q}$) groups of $E^1$ are $E^1_{1,3}$, $E^1_{1,1}$, and $E^1_{2,1}$. In this case, the differential $\partial _1: E^1_{2,1} \rightarrow E^1_{1,1}$ is an isomorphism, because the zero section of the tautological bundle over ${\mathbb {CP}}^1$ defines a non-zero element of the 2-dimensional Borel–Moore homology group of the space of this bundle. Therefore, the only surviving term is $E^2_{1,3}\,{\cong }\, {\mathbb Q}$; by Alexander duality, it gives us a $(2m-3)$-dimensional cohomology class.

The remaining statements of Theorem 3 are based on the following comparison lemma. Consider the map $J: {\mathbb C}^{d+1} \rightarrow {\mathbb C}^D$, $D=m(d+2-m),$ sending any homogeneous polynomial ${\mathbb C}^2 \rightarrow {\mathbb C}^1$ of degree d to the collection of all its partial derivatives of order $m-1$.

Lemma 7

For any $d \ge m>1$, $\Sigma _m = J^{-1}(\Sigma _{\mathbb C})$. For any $d \ge 2m-1$, the induced map of cohomology groups, $J^*: H^*({\mathbb C}^D {\setminus } \Sigma _{\mathbb C}, {\mathbb Q}) \rightarrow H^*({\mathbb C}^{d+1} {\setminus } \Sigma _m, {\mathbb Q}),$ is an isomorphism.

This is a standard comparison theorem of our spectral sequences: see especially Section IV.7 in Vassiliev (1994, 1997). $\square$

Now the assertions of Theorem 3 on the multiplication and weight filtrations follow from the similar assertions of Theorem 2 by the naturality of these structures. $\square$