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\author{V.~A.~Vassiliev\thanks{supported in part by INTAS
(project ????)
and RFBR (project ????)}}

\title{How to calculate homology groups of spaces of
nonsingular algebraic projective hypersurfaces}
\date{}

\begin{document}
\maketitle

\begin{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 $n=2, d \le 4$  and $n=3=d$ are calculated.
\end{abstract}

\section{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 \cite{book},
\cite{phasis};
it consists of two main parts. The first one is
due to Arnold, who remarked \cite{ar70}
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 {\em discriminant}
spaces of singular objects; see also \cite{ar89}. 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 \cite{viniti} and generalized in \cite{v91},
\cite{book}, \cite{phasis}.

The version of this method, applied to spaces of
nonsingular projective hypersurfaces, is described in \S~3. In the preceding
\S~2 we demonstrate all main ingredients of it, considering the
simplest problem of this series: we calculate there
the (well-known) cohomology groups of the space of regular quadrics in $\CP^2.$
In \S~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 \S~5, \S~6 and \S~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 $\C^3$.
\medskip

This work is inspired by the Arnold's problem 1970-13 on the topology of
the space of nonsingular cubics, see \cite{aprobl}.

\subsection{Stating the problem and the first reduction.}

Let $\Pi_{d,n} \simeq \C^{D(d,n)}$ be the space of all
homogeneous polynomials $\C^{n+1} \to \C^1$ of degree $d,$
$\Sigma_{d,n} \subset \Pi_{d,n}$ the set of forms, having singular
points outside the origin, and
$N_{d,n} \subset \CP^{D(d,n)-1}$  the projectivization of
$\Pi_{d,n} \setminus \Sigma_{d,n}$, i.e. the set of nonsingular
algebraic hypersurfaces of degree $d$ in $\CP^{n}.$
$\Sigma_{d,n}$  is a conical nonempty algebraic hypersurface in
$\Pi_{d,n},$ therefore
$\Pi_{d,n} \setminus \Sigma_{d,n}$ is diffeomorphic to
$N_{d,n} \times \C^*$, in particular the homology groups of
spaces $N_{d,n}$ and $\Pi_{d,n} \setminus \Sigma_{d,n}$ are related
via the K\"unneth formula.
\medskip

By the Alexander duality theorem, for $i>0$  we have
\begin{equation}
\label{alex}
\tilde H^i(\Pi_{d,n} \setminus \Sigma_{d,n}) \simeq
\bar H_{2D(d,n) -i-1}(\Sigma_{d,n}),
\end{equation}
where $\bar H_*$ is the notation for the Borel--Moore homology, i.e.
the homology of the one-point compactification reduced modulo the
added point.


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

In this section we assume that $n=2$ and denote the
spaces $\Pi_{d,2}, \Sigma_{d,2}$ and $N_{d,2}$
simply by $\Pi_{d}, \Sigma_{d}$ and $N_{d}$.
\medskip

{\sc Proposition 1.} {\em Only the following groups
$H^i(\Pi_2\setminus \Sigma_2)$ are nontrivial:}
$H^0 \simeq H^1 \simeq H^5 \simeq H^6 \simeq \Z,$
$H^3 \simeq H^4 \simeq \Z_2.$
{\em In particular, only the following groups
$H^i(N_2)$ are nontrivial:}
$H^0 \simeq H^5 \simeq \Z,$
$H^3 \simeq \Z_2.$
\medskip

This statement is not new: indeed, it is easy to see that
the space $\Pi_2 \setminus \Sigma_2$ is homotopy equivalent
to the Lagrangian Grassmannian manifold $U(3)/O(3),$
whose cohomology groups were studied in \cite{borel} and \cite{fuchs2},
see e.g. \cite{v88}.  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 $x \in \CP^2$,

B) any line $\CP^1 \subset \CP^2$ (such lines are parametrized by
the points of the dual projective space $\CP^{2\vee}$);

C) the entire $\CP^2$.

For any set of type A), B) or C), the corresponding set of
quadratic forms $\C^3 \to \C$,
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.
\medskip

\subsection{The topological order complex $\Lambda_2$}

Take the spaces $\CP^2, \CP^{2\vee}$ and $\{\bullet\}$  of sets of types
A) B) and C), and
consider the join $\CP^2 * \CP^{2\vee} * \{\bullet\} \ $
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 {\em coherent,} if
all subsets in $\CP^2$, corresponding to its vertices,
are incident to one another.

The {\em topological order complex}
$\Lambda_2$ 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 $\Lambda_2(K) \subset \Lambda_2:$ 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 $K \simeq \CP^1$, respectively, to entire $\Lambda_2$).
By ${\stackrel{\circ}{\Lambda}}_2(K)$ we denote this cone with
its base removed.

\subsection{Conical resolution}
The {\em conical resolution} $\sigma_2$
of $\Sigma_2$ is defined as a subset in the
direct product $\Pi_2 \times \Lambda_2$.  Namely, let $K\subset \CP^2$ be any
possible set of type A), B) or C).
Let $L(K)$ be the set of all quadratic forms $\C^3 \to \C$, having singular
points at all points of $K$ (and maybe somewhere else); this always is a vector
subspace in $\Pi_2.$ Then the space $\sigma_2(K) \subset \Pi_2 \times
\Lambda_2$
is defined as the direct product $L(K) \times \Lambda_2(K),$ and the desired
conical resolution $\sigma_2$ as the union of all possible sets $\sigma_2(K).$
\medskip

{\sc Proposition 2} (see \cite{book}, \cite{phasis}). {\em The obvious
projection $\sigma_2 \to \Sigma_2$ 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 $\bar H_*(\sigma_2) \simeq
\bar H_*(\Sigma_2)$.} \quad $\Box$ \medskip

The space $\sigma_2$ admits a natural filtration
$F_1 \subset F_2 \subset F_3$: the subspace $F_1$
(respectively, $F_2$, respectively, $F_3$) of
$\sigma_2$ is the union of sets $\sigma_2(K)$ 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. $F_3 = \sigma_2$).

Similarly, we consider the
filtration $\{\Phi_i\}$ of the space $\Lambda_2:$
the subspace $\Phi_1$ (respectively, $\Phi_2$,
respectively, $\Phi_3$) of $\Lambda_2$ is the union of
sets $\Lambda_2(K)$ 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. $\Phi_3 = \sigma_2$).

In particular, the space $ F_i \setminus F_{i-1}$
is the space of a complex
vector bundle over $ \Phi_i \sm \Phi_{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 \setminus F_{i-1}$ and $ \Phi_i \sm \Phi_{i-1}$
are related by the Thom isomorphism.

Consider the spectral sequence
generated by the filtration $\{F_i\}$
and converging to the group $\bar H_*(\sigma_2).$
By definition its term $E^1_{p,q}$ is equal to $\bar H_{p+q}(F_p \setminus
F_{p-1}).$
\medskip

{\sc Lemma 1.} {\em All nontrivial terms $E^1_{p,q}$
of this spectral sequence are as follows (see fig.~\ref{ss2} left):
$E^1_{1,q}=\Z$ for $q=5,7,9;$
$E^1_{2,q}=\Z$ for $q=3,5,7;$
$E^1_{3,q}=\Z$ for $q=5.$}
\medskip

\begin{figure}
\unitlength=1.00mm
\thinlines
\linethickness{0.4pt}
\begin{picture}(105.00,60.00)
\put(10.00,10.00){\vector(0,1){50.00}}
\put(10.00,10.00){\vector(1,0){40.00}}
\put(6.00,59.00){\makebox(0,0)[cc]{$q$}}
\put(6.00,52.00){\makebox(0,0)[cc]{$9$}}
\put(6.00,47.00){\makebox(0,0)[cc]{$8$}}
\put(6.00,42.00){\makebox(0,0)[cc]{$7$}}
\put(6.00,37.00){\makebox(0,0)[cc]{$6$}}
\put(6.00,32.00){\makebox(0,0)[cc]{$5$}}
\put(6.00,27.00){\makebox(0,0)[cc]{$4$}}
\put(6.00,22.00){\makebox(0,0)[cc]{$3$}}
\put(15.00,32.00){\makebox(0,0)[cc]{${\bf Z}$}}
\put(15.00,42.00){\makebox(0,0)[cc]{${\bf Z}$}}
\put(15.00,52.00){\makebox(0,0)[cc]{${\bf Z}$}}
\put(25.00,42.00){\makebox(0,0)[cc]{${\bf Z}$}}
\put(25.00,32.00){\makebox(0,0)[cc]{${\bf Z}$}}
\put(25.00,22.00){\makebox(0,0)[cc]{${\bf Z}$}}
\put(35.00,32.00){\makebox(0,0)[cc]{${\bf Z}$}}
\put(30.00,3.00){\line(0,1){54.00}}
\put(20.00,57.00){\line(0,-1){54.00}}
\put(15.00,6.00){\makebox(0,0)[cc]{$1$}}
\put(25.00,6.00){\makebox(0,0)[cc]{$2$}}
\put(35.00,6.00){\makebox(0,0)[cc]{$3$}}
\put(48.00,6.00){\makebox(0,0)[cc]{$p$}}
\put(65.00,10.00){\vector(0,1){50.00}}
\put(65.00,10.00){\vector(1,0){40.00}}
\put(61.00,59.00){\makebox(0,0)[cc]{$q$}}
\put(61.00,52.00){\makebox(0,0)[cc]{$9$}}
\put(61.00,47.00){\makebox(0,0)[cc]{$8$}}
\put(61.00,42.00){\makebox(0,0)[cc]{$7$}}
\put(61.00,37.00){\makebox(0,0)[cc]{$6$}}
\put(61.00,32.00){\makebox(0,0)[cc]{$5$}}
\put(61.00,27.00){\makebox(0,0)[cc]{$4$}}
\put(61.00,22.00){\makebox(0,0)[cc]{$3$}}
\put(70.00,32.00){\makebox(0,0)[cc]{${\bf Z}$}}
\put(70.00,52.00){\makebox(0,0)[cc]{${\bf Z}$}}
\put(80.00,22.00){\makebox(0,0)[cc]{${\bf Z}$}}
\put(85.00,3.00){\line(0,1){54.00}}
\put(75.00,57.00){\line(0,-1){54.00}}
\put(70.00,6.00){\makebox(0,0)[cc]{$1$}}
\put(80.00,6.00){\makebox(0,0)[cc]{$2$}}
\put(90.00,6.00){\makebox(0,0)[cc]{$3$}}
\put(103.00,6.00){\makebox(0,0)[cc]{$p$}}
\put(70.00,42.00){\makebox(0,0)[cc]{${\bf Z}_2$}}
\put(80.00,32.00){\makebox(0,0)[cc]{${\bf Z}_2$}}
\end{picture}
\caption{Terms $E^1$ and $E^2$ of the spectral sequence for $d=n=2$.}
\label{ss2}
\end{figure}

{\em Proof.} 1. $F_1$ is the space of a fiber bundle over
$\CP^2$ with fiber $\C^3,$ this proves the statement concerning
$E^1_{1,q}.$

2. $F_2 \setminus F_1$ is fibered over the space $\CP^{2\vee}$
of all complex lines in $\CP^2.$ The fiber over such a line $K$
is the direct product of the space $L(K) \simeq \C^1$ and the
space $\Lambda_2(K) \setminus \Phi_1.$ But $\Lambda_2(K)$ is the cone over
the set of all points of $K \sim \CP^1,$ and the base of this cone
belongs to $\Phi_1.$ Therefore $\Lambda_2(K) \setminus \Phi_1$
is an oriented 3-dimensional open disc, thus by the Thom isomorphism
$\bar H_*(F_2 \sm F_1) \simeq \bar H_{*-5}(\CP^{2\vee}),$
and the statement concerning $E^1_{2,*}$ is proved.

3. $F_3 \setminus F_2 \equiv \Phi_3 \setminus \Phi_2.$
$\Phi_2$ is the subset of the join $\CP^2 * \CP^{2\vee},$
consisting of all segments, spanning such pairs
$(x \in \CP^2,K \in \CP^{2\vee}),$ that $x \in K.$
In \cite{v91} it was proved that this subset is homeomorphic
to $S^7.$ $\Phi_3$ is the cone over $\Phi_2,$ hence
$F_3 \setminus F_2$ is an open 8-dimensional disc,
and the statement concerning $E^1_{3,*}$ also is proved.
\medskip

{\sc Proposition 3.} {\em The differential $d^1$
of our spectral sequence maps a generator of the group
$E^1_{3,5}$ into twice a generator of $E^1_{2,5},$ maps
a generator of
$E^1_{2,7}$ into twice a generator of $E^1_{1,7},$
and acts trivially on all other groups $E^1_{p,q}$.
In particular all nontrivial groups $E^2_{p,q}$
are as shown in fig.~\ref{ss2} right and our spectral sequence
degenerates at this step: $E^2 \equiv E^\infty$.}
\medskip

Proposition 1 follows immediately from this one, Lemma 1
and the Alexander isomorphism (\ref{alex}).

\medskip
{\em Proof of Proposition 3.}
1. The space $F_2 \sm F_1$ can be considered as the space of a fiber bundle
with fiber $\C^1,$ whose base $\Phi_2 \sm \Phi_1$ is fibered
over $\CP^{2\vee}$ with fiber equal to an open 3-dimensional
disc.
The image of the map
$d^1:E^1_{3,5} \to E^1_{2,5}$
is equal to the fundamental cycle
(with closed supports) of the base $\Phi_2 \setminus \Phi_1$  of this
bundle. The homomorphism
\begin{equation}
\label{thom1}
\bar H_7(\Phi_2 \sm \Phi_1) \to
\bar H_7(F_2 \sm F_1) \to
\bar H_5(\Phi_2 \sm \Phi_1),
\end{equation}
defined as the composition of the identical embedding and the
Thom isomorphism, maps this fundamental cycle into a homology class,
Poincar\'e dual to the Euler class of the complex line bundle, whose fiber
over the points of the open cone
${\stackrel{\circ}{\Lambda}}_2(K),$ $K \in \CP^{2\vee},$
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
$\bar H_5(\Phi_2 \sm \Phi_1),$ and the first statement
of Proposition 3 is proved.

2. The homomorphism
$d^1:E^1_{2,7} \to E^1_{1,7}$
maps the generator of the group $E^1_{2,7}$
into the fundamental cycle of
the algebraic subset $\omega \subset F_1$, consisting of all pairs of the form
$($a point $u \in \CP^2$, 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
$\{\alpha \beta = \gamma^2\} \subset \C^3$  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 $\CP^1 \subset \CP^2,$ say the one given by
the equation
$x=0,$ and consider any section of the restriction on it of the
fibration $F_1 \to \CP^2.$ For such a section we can take the one,
whose value over the point $(0:1:\lambda) \in \CP^1$ is the
quadratic form $\frac{x(z-\lambda y)}{1+|\lambda|^2}.$ This
section intersects the above-described set $\omega$ at exactly one point,
corresponding to $\lambda=\infty.$ This intersection is of multiplicity
2, hence the fundamental cycle of this set $\omega$ is equal to
twice the generator of $\bar H_8(F_1)$.

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

\section{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.

\subsection{Conical resolution of the discriminant:
preliminary version.}

Set $D\equiv D(d,n) =\dim \Pi_{d,n}$.
Consider all Grassmann manifolds
$G_i(\Pi_{d,n}),$ $i=1, \ldots, D,$
whose points are the complex subspaces of {\em codimension} $i$
in $\Pi_{d,n}$. Suppose that the subset $K \subset \CP^n$ is the set of
singular points of a certain homogeneous function $\C^{n+1} \to \C^1,$
and denote by $L(K)$  the subspace in $\Pi_{d,n}$ consisting of all
functions having singular points everywhere in $K$ (and maybe
somewhere else). Obviously this is a vector subspace in $\Pi_{d,n}$.
Let $\Omega_i \subset G_i(\Pi_{d,n})$ be the set of all subspaces
of the form $L(K).$ Let $\bar \Omega_i$ be the closure of
$\Omega_i$ in $G_i(\Pi_{d,n})$.

Consider the join $G_1(\Pi_{d,n}) * \cdots
* G_{D-1}(\Pi_{d,n}) * G_D(\Pi_{d,n})$
and the subset in it, consisting of all {\em coherent}
simplices of any dimensions, i.e. of such simplices,
that a) all their vertices belong to subspaces $\bar \Omega_i$,
and b) the collection of subspaces in $\Pi_{d,n}$, corresponding to all
vertices of the simplex, forms a flag.

The union of all such simplices will be denoted by $\tilde \Lambda_{d,n}$.
The {\em leading vertex} of a coherent simplex is the one
lying in the space $\bar \Omega_i$ with the greatest value of $i$.
For any point $\{L\} \in \bar \Omega_i,$ the set $\tilde \Lambda_{d,n}(L)$
is the union of coherent simplices, whose leading vertices
coincide with $L$. Obviously this is a cone with vertex $\{L\}$.
In particular, $\tilde \Lambda_{d,n} = \tilde \Lambda_{d,n}(\{\Pi_{d,n}\})$.

The subset $\tilde \sigma_{d,n}(L) \subset \Pi_{d,n} \times \tilde \Lambda_{d,n}$
is defined as the direct product $L \times \tilde \Lambda_{d,n}(L).$
The {\em conical resolution} $\tilde \sigma_{d,n}$ of $\Sigma_{d,n}$ is defined
as the union of all sets $\tilde \sigma_{d,n}(L)$ over all points $\{L\}$
of all $\bar \Omega_i,$ $i=1, \ldots, D.$
\medskip

{\sc Proposition 4.} {\em The obvious projection
$\tilde \sigma_{d,n} \to \Sigma_{d,n}$ 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
$\bar H_*(\tilde \sigma_{d,n}) \simeq \bar H_*(\Sigma_{d,n})$.}
\medskip

The space $\tilde \sigma_{d,n}$ admits a natural increasing filtration:
its term $\tilde F_i \subset \tilde \sigma_{d,n}$ is the union of
sets $\tilde \sigma_{d,n}(L)$ over all $L \in \bar \Omega_j,$
$j \le i$.

Simultaneously we consider the increasing filtration
$\{\tilde \Phi_i\}$ on the space $\tilde \Lambda_{d,n}:$
its term $\tilde \Phi_i$
is the union of sets $\tilde \Lambda_{d,n}(L)$ such that
$\mbox{codim}_{\C}L \le i.$
In particular, the space $\tilde F_i \setminus \tilde F_{i-1}$
is the space of a complex $(D-i)$-dimensional
vector bundle over $\tilde \Phi_i \setminus \tilde \Phi_{i-1},$
and their Borel--Moore homology groups are related by the Thom isomorphism.

The simplicial resolution of \S~2  is the special case of this one,
and its filtration $F_1 \subset F_2 \subset F_3$
is obtained from our present filtration $\{\tilde F_i\}$ by
renumbering its terms: $F_1=\tilde F_3 =\tilde F_4,$
$F_2=\tilde F_5,$ $F_3=\tilde F_6.$

\subsection{Reduced conical resolutions of discriminants.}

There is another, more economical conical resolution
of $\Sigma_{d,n}$, also coinciding with the previous one in the
case $d=2=n$. Namely, we can reduce all "nongeometrical" vertices
$L \in \bar \Omega_i \sm \Omega_i$ and their subordinate simplices.

\medskip
{\sc Definition.} Given a point $L \in \bar \Omega_i,$ its
{\em geometrization} $\bar L$ is the smallest plane of the form
$L(K) \in \Omega_j,$ $j \le i,$ containing $L$. In particular
$\bar L = L$ if $L \in \Omega_i.$
\medskip

{\sc Example.}  Suppose that $d \ge 3,$ $u \in \CP^n,$ and
$v(t),$ $t \in \C^1,$ is an one-parametric analytic family of
points in $\CP^n$ such that $v(t)=u \Leftrightarrow t=0.$
Then for any $t \ne 0$ the 2-point set $K(t) \equiv (u,v(t)) $
defines a subspace $L(t) \equiv L(K(t))$ of codimension $2(n+1)$ in $\Pi_{d,n},$
while the limit position $L(0)$ of these subspaces when $t \to 0$
belongs to $\bar \Omega_{2(n+1)} \sm \Omega_{2(n+1)}.$ Then
$\overline{L(0)} = L(\{u\}) \in \Omega_{n+1}.$

The segment in $\tilde \Lambda_{d,n}$,  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 $\tilde \Lambda_{d,n}$ to the (properly topologized)
union $\Lambda_{d,n}$ of all coherent simplices, whose vertices
correspond to the points of spaces $\Omega_i$ only,
but not of $\bar \Omega_i \sm \Omega_i.$ Then we lift this
reducing map to the space $\tilde \sigma_{d,n}$ and get
the desired space $\sigma_{d,n},$ with which we will work all the time.
This general reduction map is constructed as follows.

\medskip
Given a point $L \in \Omega_j,$ denote by $W(L)$ the union of all coherent
simplices in $\tilde \Lambda_{d,n}$
such that for all their vertices $L'$ their geometrizations
$\overline{L'}$  coincide with $L$. Obviously $W(L)$ is a
compact and contractible subspace of $\tilde \Lambda_{d,n}$: it
is a cone with vertex $\{L\}.$
\medskip

The {\it reducing map}
\begin{equation}
\label{redd}
red: \tilde \Lambda_{d,n} \to \Lambda_{d,n}
\end{equation}
is defined as follows. For any $i$ and any
$L \in \bar \Omega_i$ we map the point $L$ to $\bar L$
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 {\em equivalent} all points of
$\tilde \Lambda_{d,n}$
sent by this extended map into one and the same point.
The space $\Lambda_{d,n}$ is defined as the quotient space of
$\tilde \Lambda_{d,n}$ by this equivalence relation.

This factorization map defines obviously a map
$\tilde \Lambda_{d,n} \times \Pi_{d,n} \to
\Lambda_{d,n} \times \Pi_{d,n},$ and hence  also a map
\begin{equation}
\label{red}
Red: \tilde \sigma_{d,n} \to  \sigma_{d,n}
\end{equation}
of the resolved discriminant
$\tilde \sigma_{d,n} \subset \tilde
\Lambda_{d,n} \times \Pi_{d,n}$ onto some space
$\sigma_{d,n} \subset \Lambda_{d,n} \times \Pi_{d,n}$.
\medskip

{\sc Proposition 5.} {\it The map $($\ref{red}$)$
induces a homotopy equivalence of one-point compactifications of spaces
$\tilde \sigma_{d,n}$ and $ \sigma_{d,n}$ and factorizes the map
$\pi: \tilde \sigma_{d,n} \to \Sigma$
(i.e., there is a proper map
$\tilde \pi:  \sigma_{d,n} \to \Sigma$ such that
$\pi = \tilde \pi \circ Red$).}
\medskip

This follows immediately from the construction. \quad $\Box$
\medskip

For any $L \in \Omega_i$  denote by $\Lambda_{d,n}(L) $ the image
of $\tilde \Lambda_{d,n}(L)$  under the reducing map
(\ref{redd}).
\medskip

{\sc Example.} If $K$
is a set of $r < \infty$  points, and $d$ is sufficiently large
with respect to $r,$ then  both spaces $\tilde \Lambda_{d,n}(L(K))$ and
$ \Lambda_{d,n}(L(K))$ are naturally homeomorphic to a
$(r-1)$-dimensional simplex, and the restriction of the map $red$ onto
$\tilde \Lambda_{d,n}(K(L))$ is a homeomorphism between them.
\medskip

{\sc Definitions and notation.} For any set $L$ we denote by
$\partial \Lambda_{d,n}(L)$ the {\em link}  of the order complex
$\Lambda_{d,n}(L),$ i.e. the union of all its coherent simplices
not containing its vertex $\{L\}$, and denote by
$\stackrel{\circ}{\Lambda}_{d,n}(L)$ the {\em open cone}
$\Lambda_{d,n}(L) \sm \partial \Lambda_{d,n}(L).$
Their homology groups are obviously related by the boundary
isomorphism
\begin{equation}
\label{bound}
\bar H_*(\stackrel{\circ}{\Lambda}_{d,n}(L)) \cong
H_*(\Lambda_{d,n}(L),
\partial \Lambda_{d,n}(L)) \stackrel{\sim}{\to}
H_{*-1}(\partial \Lambda_{d,n}(L),pt).
\end{equation}
For any subset $K \subset \CP^n$ the complex
$\Lambda_{d,n}(L(K))$ will be usually denoted simply by
$\Lambda_{d,n}(K)$.

\subsection{Filtrations and spectral sequences.}

The space $\tilde \Lambda_{d,n}$ admits a natural increasing filtration
$\tilde \Phi_1 \subset \cdots \subset \tilde \Phi_D \equiv \tilde
\Lambda_{d,n}$.  Namely, the filtration of any interior point of a coherent
simplex $\Delta \subset \tilde \Lambda_{d,n}$ is equal to the codimension $i$
of the plane $L(\Delta) \in \bar \Omega_i$ corresponding to the leading vertex
of this simplex. Hence, a similar filtration is induced also on the
quotient space
$\Lambda_{d,n}$: the filtration of the equivalence class
$x \in \Lambda_{d,n}$ is equal to the smallest value of filtrations of points
in $\tilde \Lambda_{d,n}$, constituting $x$.  The obvious projections
$\tilde \sigma_{d,n} \to \tilde \Lambda_{d,n}$, $ \sigma_{d,n} \to \Lambda_{d,n}$
induce also filtrations on spaces
$\tilde \sigma_{d,n}$ and $ \sigma_{d,n}$.
By construction, the term
$\tilde F_i \setminus \tilde F_{i-1}$ of this filtration on
$\tilde \sigma_{d,n}$ (respectively, on $ \sigma_{d,n}$) is the space of a
$(D-i)$-dimensional complex vector bundle over the term
$\tilde \Phi_i \sm \tilde \Phi_{i-1}$
of the filtration on $\tilde \Lambda_{d,n}$
(respectively, on $\Lambda_{d,n}$).

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 $\tilde F_j$,
and its term $F_i$ is the first term $\tilde F_j$ of the former filtration
which is {\em strictly} greater than $F_{i-1}$.

This filtration defines a spectral sequence, converging to the
right-hand group in (\ref{alex}). Its term $E^1_{p,q}$ is isomorphic to
$\bar H_{p+q}(F_p \setminus F_{p-1})$.

\section{Non-singular forms of degree 3}

{\sc Theorem 1.} {\em
Only the following groups
$H^i(\Pi_{3,2}\setminus \Sigma_{3,2},\R)$ are nontrivial:}
$H^0 \simeq H^1 \simeq H^3 \simeq H^4 \simeq
H^5 \simeq H^6 \simeq H^8
\simeq H^9 \simeq \R.$
{\em In particular, only the following groups
$H^i(N_{3,2},\R)$ are nontrivial:}
$H^0 \simeq H^3 \simeq H^5 \simeq H^8 \simeq \R.$
\medskip

This theorem follows immediately from the following one.

\medskip
{\sc Theorem 2.} {\em The term $E^1$ of our spectral sequence
(over $\R$), converging to the group $\bar H_*(\Sigma_{3,2}),$
looks as in fig.~\ref{ss3} (where empty cells denote trivial
groups $E^1_{p,q}).$
$E^\infty \equiv E^1$, i.e. the unique possible differential
$d_1:E^1_{2,13} \to E^1_{1,13}$ of this sequence is trivial.}
\medskip

\begin{figure}
\unitlength=1.00mm
\thinlines
\linethickness{0.4pt}
\begin{picture}(95.00,100.00)
\put(10.00,10.00){\vector(1,0){85.00}}
\put(93.00,6.00){\makebox(0,0)[cc]{$p$}}
\put(10.00,10.00){\vector(0,1){90.00}}
\put(5.00,98.00){\makebox(0,0)[cc]{$q$}}
\put(5.00,87.00){\makebox(0,0)[cc]{$17$}}
\put(5.00,81.00){\makebox(0,0)[cc]{$16$}}
\put(5.00,75.00){\makebox(0,0)[cc]{$15$}}
\put(5.00,69.00){\makebox(0,0)[cc]{$14$}}
\put(5.00,63.00){\makebox(0,0)[cc]{$13$}}
\put(5.00,57.00){\makebox(0,0)[cc]{$12$}}
\put(5.00,51.00){\makebox(0,0)[cc]{$11$}}
\put(5.00,45.00){\makebox(0,0)[cc]{$10$}}
\put(5.00,39.00){\makebox(0,0)[cc]{$9$}}
\put(5.00,33.00){\makebox(0,0)[cc]{$8$}}
\put(5.00,27.00){\makebox(0,0)[cc]{$7$}}
\put(5.00,21.00){\makebox(0,0)[cc]{$6$}}
\put(22.00,4.00){\line(0,1){89.00}}
\put(34.00,93.00){\line(0,-1){89.00}}
\put(46.00,4.00){\line(0,1){89.00}}
\put(58.00,93.00){\line(0,-1){89.00}}
\put(70.00,4.00){\line(0,1){89.00}}
\put(16.00,87.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(16.00,75.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(16.00,63.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(28.00,63.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(28.00,51.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(28.00,39.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(52.00,21.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(64.00,6.00){\makebox(0,0)[cc]{$5_{(10)}$}}
\put(52.00,6.00){\makebox(0,0)[cc]{$4_{(9)}$}}
\put(40.00,6.00){\makebox(0,0)[cc]{$3_{(7)}$}}
\put(28.00,6.00){\makebox(0,0)[cc]{$2_{(6)}$}}
\put(16.00,6.00){\makebox(0,0)[cc]{$1_{(3)}$}}
\put(2.00,18.00){\line(1,0){69.00}}
\put(2.00,24.00){\line(1,0){69.00}}
\put(2.00,30.00){\line(1,0){69.00}}
\put(2.00,36.00){\line(1,0){69.00}}
\put(2.00,42.00){\line(1,0){69.00}}
\put(2.00,48.00){\line(1,0){69.00}}
\put(2.00,54.00){\line(1,0){69.00}}
\put(2.00,60.00){\line(1,0){69.00}}
\put(2.00,66.00){\line(1,0){69.00}}
\put(2.00,72.00){\line(1,0){69.00}}
\put(2.00,78.00){\line(1,0){69.00}}
\put(2.00,84.00){\line(1,0){69.00}}
\put(2.00,90.00){\line(1,0){69.00}}
\end{picture}
\caption{spectral sequence for $d=3$.}
\label{ss3}
\end{figure}

{\sc Remark.} Theorem 1 is not surprising: it states in fact that
$H^*(\Pi_{3,2} \sm \Sigma_{3,2}, \R) \simeq H^*(PGL(\CP^2),\R).$
There is obvious projection of the space $\Pi_{3,2} \sm \Sigma_{3,2}$
onto the modular space of elliptic curves. The group
$PGL(\CP^2)$ 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.
\medskip

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

\subsection{Main homological lemmas.}

{\sc Definition.} Given a topological space $X$, the {\em $k$-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. \cite{book}, \cite{phasis}. The {\em sign representation}
$\pi_1(B(X,k)) \to \mbox{Aut} (\Z)$ maps the
paths in $B(X,k),$  defining odd (respectively, even) permutations
of $k$ points into multiplication by $-1$ (respectively, $1$).
The local system $\pm \Z$ over $B(X,k),$ is the one
locally isomorphic to $\Z$, but with monodromy representation
equal to the sign representation of $\pi_1(B(X,k))$.
The {\em twisted Borel--Moore homology
group} $\bar H_*(B(X,k),\pm \Z)$ is defined as the
homology group of the complex of {\em locally finite} singular
chains of $B(X,k)$ with coefficients in the sheaf $\pm \Z$.
Similarly we define local systems $\pm \R \equiv \pm \Z \otimes \R$
and their homology groups.
For any topological space $X,$ $\bar X$ denotes the one-point
compactification of $X$.
\medskip

{\sc Lemma 2.} {\em A. The group
\begin{equation}
\label{cnk}
\bar H_*(B(\C^n,k), \pm \R),
\end{equation}
$n \ge 1,$ is trivial for any $k \ge 2.$

B. The group
\begin{equation}
\label{cpnk}
\bar H_*(B(\CP^n,k), \pm \R),
\end{equation}
$n \ge 1,$
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} \cite{MS}.

{\em In particular, the group (\ref{cpnk}) is trivial if $k>n+1$.}
\medskip

{\em Proof.}
Assertion A is well-known, see e.g. \cite{book}, \cite{phasis}.

Let us fix some complete flag $\bullet \subset \CP^1 \subset \CP^2 \subset
\cdots \subset \CP^n.$ To any configuration
of $k$ points in $\CP^n$ associate its
{\em index} $a=(a_0 \le a_1 \le \cdots \le a_n),$
$a_n=k,$ where $a_i$
is the number of its points lying in $\CP^i.$ For any
such index $a$ denote by $B_a \subset B(\CP^n,k)$ the
union of all configurations having this index.
It follows immediately from statement A, that
$\bar H_*(B_a,\pm \R) =0$ 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
$\bar H_*(B_a,\pm \R).$ Moreover, for any Schubert symbol $a$
the set $B_a$ is diffeomorphic to the complex vector space of dimension
$\sum_{i=1}^{n} i \cdot (a_i - a_{i-1})$. This implies statement B.
\medskip

{\sc Definition.}
For any finite-dimensional topological space $X,$
its $k$-th {\em 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 $k$-tuples 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.)
\medskip

{\sc Lemma 3.} {If $k\ge 2$, then $H_*((S^2)^{*k},\R)\equiv 0$
in all dimensions.}
\medskip

{\em Proof.} The space
$(S^2)^{*k}$
is obviously filtered
by similar self-joins:
\begin{equation}
S^1 \equiv (S^2)^{*1} \subset (S^2)^{*2} \subset \cdots
\subset (S^2)^{*(k-1)} \subset (S^2)^{*k}.
\label{sjoins}
\end{equation}
Consider the spectral sequence generated by this filtration.
Any space $\phi_i \sm \phi_{i-1} \equiv
(S^2)^{*i} \sm (S^2)^{*(i-1)}$ of this filtration
is fibered over $B(S^2,i)$
with the fiber $\stackrel{\circ}{\Delta}^{i-1}$.
This bundle is nontrivial, its fibers change their orientations
exactly over the same loops, which act nontrivially on the
local system $\pm \R$. Hence the first term of this sequence
is given by the formula
$${\cal E}^1_{p,q} \simeq
\bar H_{p+q}((S^2)^{*p} \sm (S^2)^{*(p-1)},\R) \simeq
\bar H_{q+1}(B(\CP^1,p),\pm \R).$$
By Lemma 2 there are exactly two such nontrivial cells
${\cal E}^1_{p,q}$, namely
${\cal E}^1_{2,1} \simeq {\cal E}^1_{1,1} \simeq \R$.
The unique differential $d^1$ of our spectral sequence,
which can be nontrivial, acts between these cells; it is exactly
the boundary homomorphism
$H_3((S^2)^{*2},S^2;\R) \to H_2(S^2,\R);$
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.
\medskip

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

{\sc Lemma 4.} {\em There are isomorphisms
\begin{equation}
\label{b3}
\bar H_*(B(\CP^2,3),\pm \R) \stackrel{\sim}{\longrightarrow}
\bar H_*(\tilde B(\CP^2,3),\pm \R),
\end{equation}
\begin{equation}
\label{b4}
\bar H_*(B(\CP^2,4),\pm \R) \stackrel{\sim}{\longrightarrow}
\bar H_*(\tilde B(\CP^2,4),\pm \R),
\end{equation}
induced by identical embeddings.
Namely, both groups (\ref{b4}) are trivial in all dimensions,
and groups (\ref{b3}) are trivial in all dimensions other
than 6 and are isomorphic to $\R$ in dimension 6.}
\medskip

{\em Proof.} The space
$B(\CP^2,3) \sm \tilde B(\CP^2,3)$
consists of 3-configurations lying on the same line.
This space is obviously fibered over $\CP^{2\vee}$
with fiber equal to $B(\CP^1,3).$ Applying Lemma 2 B) to
this fiber, we get the assertion about the groups (\ref{b3}).
Similarly, the space
$B(\CP^2,4) \sm \tilde B(\CP^2,4)$ is the union
of two pieces, consisting of 4-configurations,
some three  (respectively all four) points of which lie
in the same line. Both these pieces are fibered over
$\CP^{2\vee}$  with fibers $B(\CP^1,3) \times \C^2$
(respectively, $B(\CP^1,4)$). By Lemma 2 B) both these pieces
are $\pm \R$-acyclic, and Lemma 4 is proved.

\subsection{Proof of Theorem 2.}

There are exactly the following possible singular sets $K\subset \CP^2$
defined by homogeneous polynomials $\C^3 \to \C^1$ of degree 3:

A) any point $u \in \CP^2;$

B) any pair of points in $\CP^2;$ the corresponding space
$L(K)$ consists of all 3-forms 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 $\ge 2$;

D) any triple of points in $\CP^2$ not lying on the same line,
i.e. defining a point of the space $\tilde B(\CP^2,3)$.
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 $\CP^2.$
\medskip

The corresponding terms $F_i \setminus F_{i-1}$ 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, $\CP^2$, $B(\CP^2,2)$, $\CP^{2\vee},$ $\tilde B(\CP^2,3)$
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 $\stackrel{\circ}{\Lambda}_{3,2}(K)$).
The spaces of only second fiber bundles are exactly the terms
$\Phi_i \sm \Phi_{i-1}$ of the related filtration of the
continuous order complex $\Lambda_{3,2}$ of all possible sets
$K$ defined by homogeneous polynomials of order 3.

Let us describe explicitly all these terms and bundles.

A) The space $F_1 \equiv \tilde F_3$ is the space of a
fiber bundle with fiber $\C^7$ over $\CP^2$, this implies
statement of Theorem 2 concerning the column $p=1$\footnote{the
fibers of the second bundle in this case are the points}.
\medskip

B) The space $F_2 \sm F_1 \equiv \tilde F_6 \sm \tilde F_5$.
The fiber of the second bundle over a point $(u,v) \in B(\CP^2,2)$ is
the union of two segments, joining this point $(u,v)$
with both points $u$ and $v\in \CP^2$, 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 $\pm \Z$. Therefore the term $E^1_{2,q}$ of our
spectral sequence is given by the formula
\begin{equation}
\label{ss3_2}
E^1_{2,q} \cong \bar H_{2+q}(F_2 \sm F_1, \R) \simeq
\bar H_{2+q-8}(\Phi_2 \sm \Phi_1,\R) \simeq
\bar H_{2+q-9}(B(\CP^2,2),\pm \R).
\end{equation}

Thus the assertion of Theorem 2 concerning the column
$E^1_{2,q}$ of the main spectral sequence follows
from Lemma 2.
\medskip

C) Let $K=\CP^1 \subset \CP^2.$
Then the reduced order complex $\Lambda_{3,2}(K)$
is homeomorphic to the cone with vertex $\{L(K)\} \subset \Omega_7,$
and base lying in $\Phi_2.$ By the  construction, this base
$\partial \Lambda_{3,2}(K)$ is canonically homeomorphic
to the self-join $(\CP^1)^{*2},$  whose homology
groups are trivial by Lemma 3.

The space $F_3 \sm F_2$  is fibered over
$\CP^{2\vee}$; its fiber over a line $\{K\} \in \CP^{2\vee}$
is the product of the space $\C^3$ and
the space $\Lambda_{3,2}(K) \sm \Phi_2$, i.e. an open cone
with base $\partial \Lambda_{3,2}(K)$. Thus the
Borel--Moore homology group of this fiber
is acyclic over $\R,$ this implies assertion of Theorem 2 concerning
the column $p=3.$
\medskip

D) $\Phi_4 \sm \Phi_3$  is fibered
over $\tilde B(\CP^2,3);$ its fiber over a point
$(u,v,w) \subset \CP^2$ is the open triangle
$\dot \Delta^2,$ whose vertices are identified with these points
$u,v$ and $w.$ In particular,
\begin{equation}
\label{d}
\bar H_i(F_4 \sm F_3) \equiv
\bar H_{i-2}(\Phi_4 \sm \Phi_3) \simeq \bar
H_{i-4}(\tilde B(\CP^2,3),\pm \Z).
\end{equation}

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

E) Finally, the set $F_5 \sm F_4$ coincides with
$\Phi_5 \sm \Phi_4 \equiv \Lambda_{3,2} \sm \Phi_4$;
it is an open cone over the reduced topological order complex $\Phi_4,$
so that $\bar H_i(F_5 \sm F_4) \simeq \tilde H_{i-1}(\Phi_4)$.
Consider the spectral sequence $e^a_{b,c} \to H_*(\Phi_4,\R),$
generated by the filtration $\{\Phi_i\},$ $i=1,2,3,4.$
Its term $e^1_{i,c}$ is isomorphic to
$\bar H_{i+c}(\Phi_i \sm \Phi_{i-1},\R).$ The homology groups of spaces
$(\Phi_i \sm \Phi_{i-1})$ are related by the Thom isomorphism with
these of corresponding spaces $F_i \sm F_{i-1}$
(calculated in the previous steps A --- D), hence the term
$e^1$ of our sequence looks as is shown in fig.~\ref{sf5}.

\begin{figure}
\unitlength=1.00mm
\thinlines
\linethickness{0.4pt}
\begin{picture}(75.00,78.00)
\put(10.00,10.00){\vector(1,0){65.00}}
\put(72.00,6.00){\makebox(0,0)[cc]{$p$}}
\put(10.00,10.00){\vector(0,1){68.00}}
\put(5.00,76.00){\makebox(0,0)[cc]{$q$}}
\put(5.00,67.00){\makebox(0,0)[cc]{$8$}}
\put(5.00,61.00){\makebox(0,0)[cc]{$7$}}
\put(5.00,55.00){\makebox(0,0)[cc]{$6$}}
\put(5.00,49.00){\makebox(0,0)[cc]{$5$}}
\put(5.00,43.00){\makebox(0,0)[cc]{$4$}}
\put(5.00,37.00){\makebox(0,0)[cc]{$3$}}
\put(5.00,31.00){\makebox(0,0)[cc]{$2$}}
\put(5.00,25.00){\makebox(0,0)[cc]{$1$}}
\put(5.00,19.00){\makebox(0,0)[cc]{$0$}}
\put(5.00,13.00){\makebox(0,0)[cc]{$-1$}}
\put(22.00,4.00){\line(0,1){67.00}}
\put(34.00,71.00){\line(0,-1){67.00}}
\put(46.00,4.00){\line(0,1){67.00}}
\put(58.00,71.00){\line(0,-1){67.00}}
\put(16.00,37.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(16.00,25.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(16.00,13.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(28.00,49.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(28.00,37.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(28.00,25.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(52.00,43.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(52.00,6.00){\makebox(0,0)[cc]{$4$}}
\put(40.00,6.00){\makebox(0,0)[cc]{$3$}}
\put(28.00,6.00){\makebox(0,0)[cc]{$2$}}
\put(16.00,6.00){\makebox(0,0)[cc]{$1$}}
\put(2.00,16.00){\line(1,0){57.00}}
\put(2.00,22.00){\line(1,0){57.00}}
\put(2.00,28.00){\line(1,0){57.00}}
\put(2.00,34.00){\line(1,0){57.00}}
\put(2.00,40.00){\line(1,0){57.00}}
\put(2.00,46.00){\line(1,0){57.00}}
\put(2.00,52.00){\line(1,0){57.00}}
\put(2.00,58.00){\line(1,0){57.00}}
\put(2.00,64.00){\line(1,0){57.00}}
\put(2.00,70.00){\line(1,0){57.00}}
\end{picture}
\caption{spectral sequence for the term $F_5 \setminus F_4$.}
\label{sf5}
\end{figure}

\medskip
{\sc Lemma 5.} {\em The differentials
$\partial^1: e^1_{2,1} \to e^1_{1,1}$ and
$\partial^1: e^1_{2,3} \to e^1_{1,3}$ of our spectral sequence are
isomorphisms, as well as the map
$\partial^2: e^2_{4,4} \to e^2_{2,5}$.
In particular, the term $e^\infty \equiv e^3$ of our spectral sequence
consists of unique nontrivial cell $e^3_{1,-1} \simeq \R$.}
\medskip

{\em Proof.} If some of these maps is not isomorphic, then
there exists a nontrivial group
$H_{i-1}(\Phi_4,\R) \simeq \bar H_{i}(F_5 \sm F_4,\R)$
with $i<9$.
It follows from the shape of terms $E^1_{p,q}$ of the main
spectral sequence with $p \le 4$ (determined in the previous steps)
that the corresponding group $E^1_{5,i-5}$ cannot disappear
in the further steps of this sequence and thus the
group $\bar H_i(\Sigma_{3,2}) \simeq
H^{19-i}(\Pi_{3,2} \sm \Sigma_{3,2})$ also is nontrivial. This is
impossible because $\Pi_{3,2} \sm \Sigma_{3,2}$ is a 10-dimensional
Stein manifold. \quad $\Box$
\medskip

Thus we have proved the assertion of Theorem 2 concerning the shape of the term
$E^1$ of the main spectral sequence.
\medskip

{\sc Lemma 6.} {\em The homomorphism
$d^1: E^1_{2,13} \to E^1_{1,13}$ is trivial.}
\medskip

{\em Proof.} It is clear from fig.~\ref{ss3}, that the
group $E_{1,15}$ cannot disappear,
hence $\bar H_{16}(\Sigma_{3,2},\R) \simeq
H^3(\Pi_{3,2} \sm \Sigma_{3,2},\R) \sim
\R.$ $\Pi_{3,2} \sm \Sigma_{3,2}$ is a direct product of something and $\C^*$,
therefore the
Poincar\'e polynomial of $ H^*(\Pi_{3,2} \sm \Sigma_{3,2},\R)$
is divisible by $(1+t).$ Since $\bar H_{17}(\Sigma_{3,2},\R) \simeq
H^2(\Pi_{3,2} \sm \Sigma_{3,2},\R)=0,$ this implies
that $\bar H_{15}(\Sigma_{3,2},\R) \ne 0.$ But the unique cell $E^1_{p,q}$ with
$p+q=15$ is $E^1_{2,13},$ and our lemma is proved.
\medskip

It terminates the proof of Theorems 2 and 1.

\section{Non-singular quartics in $\CP^2$}

{\sc Theorem 3.} {\em The Poincar\'e polynomial of the
real cohomology group of the space
$N_{4,2}$ of nonsingular curves of degree 4 in $\CP^2$
is equal to $(1+t^3)(1+t^5)(1+t^6)$.}
\medskip

For the proof, consider the standard conical resolution of
$\Sigma_{4,2}$ described in \S~3. To describe it explicitly,
we must classify all singular sets in $\CP^2$
of polynomials $\C^3 \to \C^1$ of degree 4.
\medskip

{\sc Proposition 6.} {\em 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):

\begin{enumerate}
\item
Any point in $\CP^2$
\quad $\langle 12 \rangle$

\item
Any pair of points
\quad $\langle 9 \rangle$

\item
Any three point on the same line
\quad $\langle 7 \rangle$
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\item
Any generic triple of points (i.e. not lying on the same line)
\quad $\langle 6 \rangle$

\item
Any line $\CP^1 \subset \CP^2$
\quad $\langle 6 \rangle$

\item
Any three points on the same line plus a point not on the line
\quad $\langle 4 \rangle$
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\item
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)
\quad $\langle 3 \rangle$

\item
Any line $\CP^1 \subset \CP^2$ plus a point outside it
\quad $\langle 3 \rangle$

\item
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)
\quad $\langle 2 \rangle$
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\item
Six points, which are all the intersection points of some four
lines in general position
\quad $\langle 1 \rangle$
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\item
Any nonsingular quadric in $\CP^2$
\quad $\langle 1 \rangle$

\item
Two lines in $\CP^2$
\quad $\langle 1 \rangle$

\item
Entire $\CP^2$.
\quad $\langle 0 \rangle$ \quad $\Box$
\end{enumerate}
}

I have  liberty to improve slightly the canonical filtration in the
resolution space $\sigma_{4,2}$ of the discriminant and in the underlying
order complex $\Lambda_{4,2}$  of singular sets. Namely,
for any $i=1, \ldots, 13,$ I
denote by $F_i$ (respectively, by $\Phi_i$) the union of all sets
$\sigma_{4,2}(K) \subset \sigma_{4,2}$
(respectively, $\Lambda_{4,2}(K) \subset
\Lambda_{4,2}$) where $K$ is one of sets described in items $1, \ldots, i$ of
the previous proposition.  \medskip

{\sc Proposition 7.} {\em The term $E^1$ of the spectral sequence,
generated by this filtration and converging to the group
$\bar H_*(\sigma_{4,2}, \R),$ is as shown in Fig.~\ref{ssquart}.
This spectral sequence degenerates at this term:
$E^\infty \equiv E^1$.}
\medskip


\begin{figure}
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\put(5.00,96.00){\makebox(0,0)[cc]{17}}
\put(5.00,102.00){\makebox(0,0)[cc]{18}}
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\put(5.00,114.00){\makebox(0,0)[cc]{20}}
\put(5.00,120.00){\makebox(0,0)[cc]{21}}
\put(5.00,126.00){\makebox(0,0)[cc]{22}}
\put(5.00,132.00){\makebox(0,0)[cc]{23}}
\put(5.00,138.00){\makebox(0,0)[cc]{24}}
\put(5.00,144.00){\makebox(0,0)[cc]{25}}
\put(5.00,150.00){\makebox(0,0)[cc]{26}}
\put(5.00,156.00){\makebox(0,0)[cc]{27}}
\put(6.00,162.00){\makebox(0,0)[cc]{$q$}}
\put(118.00,-5.00){\makebox(0,0)[cc]{$p$}}
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\put(10.00,147.00){\line(1,0){105.00}}
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\put(10.00,159.00){\line(1,0){105.00}}
\put(14.00,156.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(14.00,144.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(14.00,132.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(22.00,132.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(22.00,120.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(22.00,108.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(38.00,90.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(5.00,84.00){\makebox(0,0)[cc]{15}}
\put(5.00,78.00){\makebox(0,0)[cc]{14}}
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\put(10.00,21.00){\line(1,0){105.00}}
\put(86.00,72.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(86.00,54.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(86.00,42.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(86.00,24.00){\makebox(0,0)[cc]{${\bf R}$}}
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\put(110.00,18.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(110.00,0.00){\makebox(0,0)[cc]{${\bf R}$}}
\end{picture}
\caption{Spectral sequence for the space of nonsingular homogeneous
polynomials of degree 4 in $\C^3$}
\label{ssquart}
\end{figure}

{\em Proof.} The shape of columns
$p={\bf 1,2}$ and ${\bf 4}$ of $E^1$ is justified in
essentially the same way as that of
the same columns of the spectral sequence presented in
Fig.~\ref{ss3}. The triviality of columns $p={\bf 3}$ and $p={\bf 6}$ follows
from Lemma 2 ($n=1,$ $k=3$), and that of the column $p={\bf 7}$ from the same
lemma and additionally Lemma 4.
For any set $K$ of type ${\bf 5}$  (i.e., $K = \CP^1$) the link
$\partial (\Lambda_{4,2}(K))$  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 ${\bf 11}$  (i.e., $K$ is a generic
quadric in $\CP^2$) the link
$\partial (\Lambda_{4,2}(K))$
is the space $K^{*4}$, and the triviality of the column
$p=11$ also follows from Lemma 3.

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

In the case mentioned in the column $p={\bf 10}$ the
term $F_{10} \sm F_9$ is the space of a fiber bundle, whose base
is the configuration space $\tilde B(\CP^{2\vee},4)$ 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 {\em orientable},
thus $\bar H_*(F_{10}\sm F_9,\R) \simeq
\bar H_{*-7}(\tilde B(\CP^{2\vee},4),\R).$ Let us calculate the
latter homology group. By the projective duality,
$\tilde B(\CP^{2\vee},4)$
is diffeomorphic to the space
$\tilde B(\CP^{2},4)$ of all generic collections of 4 points,
see e.g. Lemma 4. The similar space
$\tilde {\cal F}(\CP^2,4)$  of {\em ordered} generic collections of points
is obviously diffeomorphic to the group $PGL(\CP^2),$ 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
$\tilde {\cal F}(\CP^2,4) \stackrel{S(4)}{\longrightarrow} \tilde B(\CP^2,4)$,
and the shape of the column $p=10$ follows.

The triviality of the column $p={\bf 8}$ follows from the
following lemma.
\medskip

{\sc Lemma 7.} {\em For any set $K \subset \CP^2$ of type 8,
$K =\CP^1 \sqcup *$, the link $\partial \Lambda_{4,2}(K)$ of the
corresponding order complex $\Lambda_{4,2}(K)$ is homeomorphic to
the suspension $\Sigma(\partial \Lambda_{4,2}(\CP^1))$ of the
similar link for the set $\CP^1$ of type 5.}
\medskip

{\em Proof.} This link consists of two pieces, the first of which is
the order complex $\Lambda_{4,2}(\CP^1)$ (obviously homeomorphic to
the cone over its own link), and the second is the union of
all order complexes $\Lambda_{4,2}(\kappa),$ where $\kappa$  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 $* = K \sm \CP^1,$
and another three are arbitrary different points of $\CP^1$.
Thus the second piece also is homeomorphic to the cone over
$\partial \Lambda_{4,2} \sim (S^2)^{*3}$. The intersection
of these two cones is their common base, and Lemma 7 is proved.
\medskip

Now let $K$ be of type ${\bf 12}$,  i.e. the union of two complex lines
$l_1, l_2 \subset \CP^2.$ The corresponding link $\partial \Lambda_{4,2}(K)$
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 $\Lambda_{4,2}(\kappa),$
where $\kappa$ is the set of type 8 consisting of the line $l_i$
and a point of the other line $K \sm l_i.$ The triviality of the
column $p=12$ follows now from the Mayer--Vietoris formula
and the following lemma.
\medskip

{\sc Lemma 8.} {\em All three complexes $A_1$, $A_2$ and $A_1 \cap A_2$
are acyclic (over $\R$) in all positive dimensions.}
\medskip

{\em Proof.} Consider the following filtration of the set
$A_1$:
$$\Lambda_{4,2}(l_1) \subset A_1 \cap \Phi_6 \subset A_1 \cap \Phi_8
\equiv A_1.$$
The set $A_1 \sm (A_1 \cap \Phi_6)$ is a fiber bundle, whose base
is the space $(K \sm l_1) \sim \C^1$, parametrizing all possible sets of type
8 containing $l_1,$ and fiber $\stackrel{\circ}{\Lambda}_{4,2}(\kappa),$
where $\kappa$ is such a set. By Lemma 7 the Borel--Moore homology group
of this difference (or, which is the same, the
relative group $H_*(A_1, (A_1 \cap \Phi_6);\R)$) is trivial.

The space $(A_1 \cap \Phi_6) \sm \Lambda_{4,2}(l_1)$
also is a fiber bundle with base $(K \sm l_1) \sim \C^1$; its
fiber is the space $\partial \Lambda_{4,2}(l_1) \sim (S^2)^{*3}$.
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 $ \Lambda_{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, $A_1 \cap A_2$ 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 $l_1 \cap l_2$);

c) any triple of points in $K$, the first of which is the
crossing point $l_1 \cap l_2$, 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. \medskip

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.~\ref{ssquart},
coincides with the stable term $E^\infty,$ 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. \quad $\Box$

\section{Non-singular cubical surfaces in $\CP^3$}

{\sc Theorem 4.} {\em The Poincar\'e 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 $H^*(N_{3,3},\R) \simeq H^*(PGL(\CP^3),\R).$}
\medskip

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

\begin{enumerate}
\item
Any point in $\CP^3$
\quad $\langle 16 \rangle$

\item
Any pair of points
\quad $\langle 12 \rangle$

\item
Any line
\quad $\langle 10 \rangle$

\item
Any triple of points in general position (i.e. not on the same line)
\quad $\langle 8 \rangle$

\item \label{plquad}
Any generic quadric inside any
plane $\CP^2 \subset \CP^3$
\quad $\langle 5 \rangle$

\item
Any pair of intersecting lines in $\CP^3$
\quad $\langle 5 \rangle$

\item
Any generic quadruple of points (i.e. not lying in the same
two-dimensional plane)
\quad $\langle 4 \rangle$

\item Any plane $\CP^2 \subset \CP^3$
\quad $\langle 4 \rangle$

\item
Any triple of lines having one common point (such a polynomial
is equal to $F= xyz$ in appropriate coordinates)
\quad $\langle 1 \rangle$

\item
Any generic plane quadric (as in item \ref{plquad}) plus any point not
in the same plane
\quad $\langle 1 \rangle$

\item
Entire $\CP^3.$
\quad $\langle 0 \rangle$
\end{enumerate}
}

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\put(50.00,5.00){\line(0,1){133.00}}
\put(58.00,5.00){\line(0,1){133.00}}
\put(66.00,5.00){\line(0,1){133.00}}
\put(74.00,5.00){\line(0,1){133.00}}
\put(82.00,5.00){\line(0,1){133.00}}
\put(90.00,5.00){\line(0,1){133.00}}
\put(98.00,5.00){\line(0,1){133.00}}
\put(14.00,2.00){\makebox(0,0)[cc]{1}}
\put(22.00,2.00){\makebox(0,0)[cc]{2}}
\put(30.00,2.00){\makebox(0,0)[cc]{3}}
\put(38.00,2.00){\makebox(0,0)[cc]{4}}
\put(46.00,2.00){\makebox(0,0)[cc]{5}}
\put(54.00,2.00){\makebox(0,0)[cc]{6}}
\put(62.00,2.00){\makebox(0,0)[cc]{7}}
\put(70.00,2.00){\makebox(0,0)[cc]{8}}
\put(78.00,2.00){\makebox(0,0)[cc]{9}}
\put(86.00,2.00){\makebox(0,0)[cc]{10}}
\put(94.00,2.00){\makebox(0,0)[cc]{11}}
\put(14.00,134.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(14.00,122.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(14.00,110.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(14.00,98.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(38.00,68.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(38.00,56.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(38.00,44.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(38.00,32.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(22.00,110.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(22.00,98.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(22.00,74.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(22.00,62.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(22.00,86.00){\makebox(0,0)[cc]{${\bf R}^2$}}
\put(62.00,8.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(115.00,2.00){\makebox(0,0)[cc]{$p$}}
\put(5,142.00){\makebox(0,0)[cc]{$q$}}
\end{picture}
\caption{Spectral sequence for nonsingular cubical
polynomials in $\C^4$}
\label{sssurf}
\end{figure}

Exactly as in the proof of Theorem 3,
for any $i=1, \ldots, 11,$ denote by $F_i$ the union of
all sets $\sigma_{3,3}(K) \subset
\sigma_{3,3}$ where $K$ is one of sets described in items
$1, \ldots, i$ of the previous proposition.
\medskip

{\sc Proposition 9.} {\em The term $E^1$ of the spectral sequence,
generated by this filtration and converging to the group
$\bar H_*(\sigma_{3,3}, \R),$ is as shown in Fig.~\ref{sssurf}.
This spectral sequence degenerates at this term:
$E^\infty \equiv E^1$.}
\medskip

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

Denote by $\tilde B(\CP^3,k)$ the subset in $B(\CP^3,k),$
consisting of {\em 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.
\medskip

{\sc Lemma 9.} {\em There are isomorphisms
\begin{equation}
\label{b33}
\bar H_*(B(\CP^3,3),\pm \R) \stackrel{\sim}{\longrightarrow}
\bar H_*(\tilde B(\CP^3,3),\pm \R),
\end{equation}
\begin{equation}
\label{b43}
\bar H_*(B(\CP^3,4),\pm \R) \stackrel{\sim}{\longrightarrow}
\bar H_*(\tilde B(\CP^3,4),\pm \R),
\end{equation}
induced by identical embeddings.
Namely, both groups (\ref{b33}) are isomorphic to $\R$ in
dimensions 6,8,10 and 12, and are
trivial in all other dimensions,
and groups (\ref{b43}) are trivial in all dimensions other
than 12 and are isomorphic to $\R$ in dimension 12.}
\medskip

{\em Proof.} The isomorphism (\ref{b33}) follows from exactly the same
arguments as (\ref{b3}), only with space $\CP^{2\vee}$ of all
lines in $\CP^2$  replaced by the space $G_2(\C^4)$ of
lines in $\CP^3$.

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

For $K$ of type ${\bf 3}$, $K=\CP^1 \subset \CP^2,$
$\partial \Lambda_{3,3}(K) \simeq (S^2)^{*2},$  therefore
the shape of the column $p={\bf 3}$ follows from Lemma 3.

Similarly, for $K$ of type ${\bf 5}$
$\partial \Lambda_{3,3}(K) \simeq (S^2)^{*3},$ and the shape of
the column $p=5$ follows from the same lemma.

For $K$ of type ${\bf 6}$,  $K=l_1 \cup l_2,$ the link
$\partial \Lambda_{3,3}(K)$
is covered by three subsets:
$A_1=\Lambda_{3,3}(l_1),$
$A_2=\Lambda_{3,3}(l_2),$
and $A_3$ swept out by all order complexes $\Lambda_{3,3}(\kappa),$
where $\kappa$ are 3-point sets, one whose point is the crossing point
$l_1 \cap l_2,$ 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
$H_*(A_1 \cup A_2) \simeq H_*(pt).$ Further,
$A_3 \sm (A_1 \cup A_2)$  is a fiber bundle, whose base is the space
$\C^1 \times \C^1 \equiv (K \sm l_1) \times (K \sm l_2)$ parametrizing
all above-described sets $\kappa$, 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 K\"unneth formula
$$\bar H_*(\partial \Lambda_{3,3}(K) \sm (A_1 \cup A_2)) \equiv
H_*(\partial \Lambda_{3,3}(K), (A_1 \cup A_2)) \equiv 0.$$
This proves the triviality of the column $p=6.$

Similarly, if $K$ is of type ${\bf 9},$
$K = l_1 \cap l_2 \cap l_3,$ then the link
$\partial \Lambda_{3,3}(K)$
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
$\Lambda_{3,3}(\mu_i),$  $\mu_i$ are unions of some two
of three lines constituting $K$. $A_4$ is swept out by
 all order complexes $\Lambda_{3,3}(\kappa),$
where $\kappa$ 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 $l_3$.
$A_4 \sm (A_1 \cup A_2\cup A_3)$  is a fiber bundle, whose
base is the space
$\C^1 \times \C^1 \times \C^1$
parametrizing all above-described sets $\kappa$, 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 K\"unneth formula
$$\bar H_*(\partial \Lambda_{3,3}(K) \sm (A_1 \cup A_2 \cup A_3)) \equiv
H_*(\partial \Lambda_{3,3}(K), (A_1 \cup A_2 \cup A_3)) \equiv 0.$$
Further, the order complexes $A_1,$ $A_2$ and $A_3$ are acyclic,
their pairwise intersections are the order complexes
$\Lambda_{3,3}(l_j),$ which also are acyclic, and their total
intersection is a point. This proves the triviality of the
column $p=9.$

For $K$ of type ${\bf 10},$ (i.e. consisting of a generic plane quadric
$\tilde K$ and one point not in the same plane), the link
$\partial \Lambda_{3,3}(K)$
is homeomorphic to the suspension
$\Sigma (\partial \Lambda_{3,3}(\tilde K)),$ cf. Lemma 7.
This proves the triviality of the column $p=10.$

{\em Column} $p={\bf 8}$. Let $K = \CP^2 \subset \CP^3.$
The link
$\partial \Lambda_{3,3}(K)$ is filtered by its intersections
with terms  $\Phi_i$ of the filtration of $\Lambda_{3,3}.$
We need only the following segment of this filtration:
\begin{equation}
\label{filtr}
\partial \Lambda_{3,3}(K) \cap \Phi_4 \subset
\partial \Lambda_{3,3}(K) \cap \Phi_5 \subset
\partial \Lambda_{3,3}(K) \cap \Phi_6 \equiv
\partial \Lambda_{3,3}(K).
\end{equation}
The first term of this filtration,
$\partial \Lambda_{3,3}(K) \cap \Phi_4,$
is acyclic by Lemma 5. The set
$(\partial \Lambda_{3,3}(K) \cap \Phi_5) \sm
(\partial \Lambda_{3,3}(K) \cap \Phi_4)$
is the space of a fiber bundle, whose base is the space of all
generic quadrics in $\CP^2$ (studied in \S~2), and the fiber over such a
quadric $\kappa$ is the open cone
$\stackrel{\circ}{\Lambda}_{3,3}(\kappa).$ 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
$\partial \Lambda_{3,3}(\kappa) \sim (S^2)^{*3},$ which is
trivial by Lemma 3. Finally, the set
$(\partial \Lambda_{3,3}(K) \cap \Phi_6) \sm
(\partial \Lambda_{3,3}(K) \cap \Phi_5)$ is the space of a fiber bundle,
whose base is the space $B(\CP^{2\vee},2)$ of pairs of complex lines in
$\CP^2$, and the fiber over such a pair $\kappa$ is the open cone
$\stackrel{\circ}{\Lambda}_{3,3}(\kappa).$ By the boundary
isomorphism (\ref{bound}), to find the Borel--Moore homology of
such a fiber, we need only to calculate the (usual)
homology of the link $\partial \Lambda_{3,3}(\kappa),$
which already was calculated (see the study of the column $p=6$)
and is trivial.

This proves the triviality of the column $p={\bf 8}$ and terminates
the justification of columns 1 through 10 of Fig.~\ref{sssurf}.
The triviality of the column $p={\bf 11}$ 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 $\Pi_{3,3} \sm \Sigma_{3,3}$ 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.

\section{Non-degenerate quadratic vector fields in $\C^3$}

{\sc Definition.} A collection of three homogeneous functions
$(v_1,v_2,v_3):\C^3 \to \C^3$ is nondegenerate, if they vanish
simultaneously only at the origin of the issue space.
\medskip

{\sc Theorem 5.} {\em The Poincar\'e polynomial of the
rational cohomology group of the space
of nondegenerate triples of homogeneous functions of
degrees 2 in $\C^3$ is equal to $(1+t)(1+t^3)(1+t^5)$.}
\medskip

{\em Proof.} Denote by
$\Sigma(2,2,2)$ 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
$\Sigma_{d,n}$ described in \S~3; instead of possible singular
subsets in $\CP^n$ we use possible sets of common zeros of a triple of
polynomials.
\medskip

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

\begin{enumerate}
\item
Any point in $\CP^2$
\quad $\langle 15 \rangle$

\item
Any pair of points
\quad $\langle 12 \rangle$

\item
Any generic triple of points (i.e. not lying on the same line)
\quad $\langle 9 \rangle$

\item
Any line $\CP^1 \subset \CP^2$
\quad $\langle 9 \rangle$

\item
Any generic quadruple of points
\quad $\langle 6 \rangle$

\item
Any line $\CP^1 \subset \CP^2$ plus a point outside it
\quad $\langle 6 \rangle$

\item
Any nonsingular quadric in $\CP^2$
\quad $\langle 3 \rangle$

\item
Two lines in $\CP^2$
\quad $\langle 3 \rangle$

\item
Entire $\CP^2$.
\quad $\langle 0 \rangle$ \quad $\Box$
\end{enumerate}
}
\medskip

{\sc Proposition 11.} {\em The term $E^1$ of the spectral sequence,
generated by the corresponding filtration and converging to the group
$\bar H_*(\Sigma_(2,2,2) \R),$ is as shown in Fig.~\ref{ssvect}.
This spectral sequence degenerates at this term:
$E^\infty \equiv E^1$.}
\medskip

\begin{figure}
\unitlength=1.00mm
\thinlines
\linethickness{0.4pt}
\begin{picture}(95.00,87.00)
\put(2.00,10.00){\vector(1,0){93.00}}
\put(93.00,6.00){\makebox(0,0)[cc]{$p$}}
\put(10.00,4.00){\vector(0,1){83.00}}
\put(5.00,85.00){\makebox(0,0)[cc]{$q$}}
\put(18.00,4.00){\line(0,1){77.00}}
\put(26.00,81.00){\line(0,-1){77.00}}
\put(34.00,4.00){\line(0,1){77.00}}
\put(42.00,81.00){\line(0,-1){77.00}}
\put(50.00,4.00){\line(0,1){77.00}}
\put(14.00,73.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(14.00,61.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(14.00,49.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(22.00,49.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(22.00,37.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(22.00,25.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(30.00,13.00){\makebox(0,0)[cc]{${\bf R}$}}
\put(46.00,6.00){\makebox(0,0)[cc]{$5$}}
\put(38.00,6.00){\makebox(0,0)[cc]{$4$}}
\put(30.00,6.00){\makebox(0,0)[cc]{$3$}}
\put(22.00,6.00){\makebox(0,0)[cc]{$2$}}
\put(14.00,6.00){\makebox(0,0)[cc]{$1$}}
\put(2.00,16.00){\line(1,0){82.00}}
\put(2.00,22.00){\line(1,0){82.00}}
\put(2.00,28.00){\line(1,0){82.00}}
\put(2.00,34.00){\line(1,0){82.00}}
\put(2.00,40.00){\line(1,0){82.00}}
\put(2.00,46.00){\line(1,0){82.00}}
\put(2.00,52.00){\line(1,0){82.00}}
\put(2.00,58.00){\line(1,0){82.00}}
\put(2.00,64.00){\line(1,0){82.00}}
\put(2.00,70.00){\line(1,0){82.00}}
\put(2.00,76.00){\line(1,0){82.00}}
\put(5.00,73.00){\makebox(0,0)[cc]{33}}
\put(5.00,67.00){\makebox(0,0)[cc]{32}}
\put(5.00,61.00){\makebox(0,0)[cc]{31}}
\put(5.00,55.00){\makebox(0,0)[cc]{30}}
\put(5.00,49.00){\makebox(0,0)[cc]{29}}
\put(5.00,43.00){\makebox(0,0)[cc]{28}}
\put(5.00,37.00){\makebox(0,0)[cc]{27}}
\put(5.00,31.00){\makebox(0,0)[cc]{26}}
\put(5.00,25.00){\makebox(0,0)[cc]{25}}
\put(5.00,19.00){\makebox(0,0)[cc]{24}}
\put(5.00,13.00){\makebox(0,0)[cc]{23}}
\put(58.00,81.00){\line(0,-1){77.00}}
\put(66.00,4.00){\line(0,1){77.00}}
\put(74.00,81.00){\line(0,-1){77.00}}
\put(82.00,4.00){\line(0,1){77.00}}
\put(54.00,6.00){\makebox(0,0)[cc]{6}}
\put(62.00,6.00){\makebox(0,0)[cc]{7}}
\put(70.00,6.00){\makebox(0,0)[cc]{8}}
\put(78.00,6.00){\makebox(0,0)[cc]{9}}
\end{picture}
\caption{Spectral sequence for nondegenerate quadratic
vector fields in $\C^3$}
\label{ssvect}
\end{figure}


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. \ref{ss3} and \ref{ssvect} implies the following version of the
(homological) Smale--Hirsch principle.
\medskip

{\sc Proposition 12.} {\em The gradient embedding
$\C^{10} \to \C^{18}$, mapping
any homogeneous polynomial $\C^3 \to \C^1$ 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 $\C^3$.} \quad $\Box$

\section{Problems and concluding remarks}

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

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 $H^*(N_{3,n},\R) \simeq H^*(PGL(\CP^n),\R)$ for any $n$?
\medskip

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 \cite{har} considered the topology of discriminant
sets in a connection with the rigid classification problem.
\medskip

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

4. A collection of points in $\RP^2$ or in $\CP^2$ is called
$d$-{\em sufficient} (respectively, $d$-{\em 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.  \medskip

{\sc Problem.} To study the set of such configurations.
For which smallest $d$ there are {\it 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 $d$-nondegenerate
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 $d$-{\em overdeterminacy} property:
any homogeneous polynomial of degree $d$,
having singular points at this subset,
is identically equal to 0.
\medskip

More generally, let us call a point
$y \in \PP^2$ \ a $d$-{\em consequence} of points $x_1, \ldots, x_l$,
if any homogeneous polynomial of degree $d$,
having singularities at these  $l$ points, has also a singularity at
the point $y$. E.g., if $(x_1, \ldots,
x_l)$ 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?
\medskip

Consider some natural family of algebraic subsets in
$\PP^2$, 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 $\Pi_d$: 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''?

\begin{thebibliography}{99}

\bibitem{ar70} V.~I.~Arnold, {\it On some topological invariants
of algebraic functions}. Transact. Moscow Math. Society
%О некоторых топологических инвариантах
%алгебраических функций}. Труды Моск. Матем. Общества
1970 {\bf 21}, 27--46.


\bibitem{ar89} V.~I.~Arnold, {\it
Spaces of functions with mild singularities.}
Funct. Anal. and its Appl.
%Пространства функций с умеренными
%особенностями}.  Функц. анализ и его прил.
1989 {\bf 23} (3), 1--10.


\bibitem{aprobl} V.~I.~Arnold, {\em Problems},
Moscow: Phasis, 1999 (in preparation).

\bibitem{borel} A.~Borel. {\it Sur la cohomologie des espaces
fibr\'{e}s principaux et des espaces homog\`{e}nes de groupes de Lie
compacts.} Ann. of Math. 1953 {\bf 57} (2), 115--207.
%Имеется
%перевод: Борель~А., {\em О когомологиях главных расслоенных
%пространств и однородных пространств компактных групп Ли}. В
%кн.: Расслоенные пространства и их приложения: Сб. переводов.
%М.: Изд-во иностранной литературы, 1958, 163--246.

\bibitem{fuchs} D.~B.~Fuchs. {\it
Cohomology of the braid group
%Когомологии группы кос
$\mathop{\rm mod} 2$.}
%Функц. анализ и его прил.
Funct. Anal. and its Appl.
1970 {\bf 4}
(2), 62--73.

\bibitem{fuchs2} D.~B.~Fuchs, {\it Maslov--Arnold characteristic
classes}, Doklady (C.R.) of Soviet Ac.Sci., 1968, {\bf 178}, 303--306.

\bibitem{har} V.~M.~Kharlamov, {\it Rigid isotopy classification
of real plane curves of degree 5},
Funct. Anal. and its Appl., 1981, 15:1.

\bibitem{MS} J.~W.~Milnor, J.~D.~Stasheff, {\it Characteristic
Classes,} Princeton, NJ -- Tokyo: Princeton Univ. Press, Univ. of
Tokyo Press, 1974, 331 pp. (Ann. of Math. Studies, 76).
%Пер. на рус. яз.: Дж.~Милнор, Дж.~Сташеф, {\it Характеристические
%классы,} М.: Мир, 1979, 371 с.

\bibitem{viniti} V.A.Vassiliev, {\it Stable cohomology of complements of
discriminants of singularities of smooth functions.
%Стабильные когомологии дополнений к дискриминантам
%особенностей гладких функций.
}
Current Problems of
Math., Newest Results, vol. 33, Itogi Nauki i Tekhniki, VINITI, Moscow, 1988,
3--29; (Russian,
English transl. in  J. Soviet Math. 52:4 (1990), 3217--3230.)

\bibitem{v88} V.~A.~Vassiliev. {\it Lagrange and Legendre
characteristic classes.} 2nd edition. New York a.o.: Gordon \&
Breach Science Publishers, 1993. 265~pp.

\bibitem{v91} V.~A.~Vassiliev, {\it
A geometric realization of the homology of classical
Lie  groups,  and  complexes, $S$-dual to the flag manifolds,}
%Геометрическая реализация гомологий классических групп Ли
%и комплексы, $S$-двойственные к флаговым многообразиям}
Algebra i Analiz 3:4 (1991), 113--120. (Russian, English transl. in
St.-Petersburg Math. J., 3:4 (1992), 809--815.)

\bibitem{book} V.~A.~Vassiliev, {\it Complements of discriminants of smooth
maps: topology and applications. Revised ed.},
Providence RI: AMS,  1994. (Transl. of Math. Monographs, 98)

\bibitem{congress} V.~A.~Vassiliev, {\it Topology of discriminants
and their complements,}
Proceedings of the Intern. Congress of Math. (Z\"urich, 1994);
Birkh\"auser Verlag, Basel; 1995, 209--226.

\bibitem{phasis} V.~A.~Vassiliev, {\em Topology of complements of
discriminants,} Moscow: Phasis, 1997 (in Russian).

%\bibitem{vera} Vassiliev, Serganova,
%On the number of real and complex moduli of singularities of
%smooth functions and realizations of maroids, Matematicheskie
%Zametki (Math. Notes), 1991, 49:1.

\end{thebibliography}

\end{document}
