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\title{Characteristic classes of multisingularities,\\
cobordisms, and enumerative geometry}
\author{Maxim Kazarian\\\em kazarian@mccme.ru}
\date{}
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\def\({\ifmmode\left(\else{\rm(}\fi}
\def\){\ifmmode\right)\else{\rm)}\fi}
\def\ot{\longleftarrow}
\def\pt{{\rm pt}}
\def\Id{{\rm Id}}
\def\Hom{\mathop{\rm Hom}}
\def\Aut{\mathop{\rm Aut}}
\def\codim{\mathop{\rm codim}}
\def\rk{\mathop{\rm rk}}
\def\Sym{\mathop{\rm Sym}\nolimits}

\def\a{\alpha}
\def\b{\beta}
\def\g{\gamma}
\def\d{\delta}
\def\l{\lambda}
\def\s{\sigma}
\def\t{\tau}

\def\R{{\mathbb R}}
\def\Z{{\mathbb Z}}
\def\Q{{\mathbb Q}}
\def\C{{\mathbb C}}
\def\G{\Gamma}
\def\O{\Omega}
\def\S{\Sigma}

\def\fm{{\mathfrak m}}
\def\cO{{\cal O}}
\def\cK{{\cal K}}
\def\cL{{\cal L}}
\def\cC{{\cal C}}
\def\cSC{{\cal SC}}
\def\cSL{{\cal SL}}
\def\om{\overline m}
\def\on{\overline n}
\def\ua{{\underline\alpha}}
\def\tM{{\widetilde M}}
\def\tW{{\widetilde W}}
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\tableofcontents


\section{Introduction}

Enumerative geometry is traditionally considered as a
subject of intersection theory. Modern intersection theory with its
methods of singular schemes, blowing-ups, residue intersections, and
Hilbert schemes is rather technical and a rigorous proof of very
simple geometrically evident results require often many pages of
complicated formulas and commutative diagrams.

For example, the number of planes in $\C P^3$ touching a fixed degree
$d$ smooth hypersurface at three points was computed by Salmon in the
middle of XIX-th century but the proof that would satisfy
modern algebraists has appeared comparatively recently. On the other
hand, the answer to the similar question about hypersurfaces in $\C P^n$,
for $n>3$ seems to be unknown.

In this note we present an alternative approach to enumerative
geometry whose motivation belongs to singularity theory and topology,
more precisely, to the theory of cobordisms and cohomological
operations. We announce a universal formula that expresses the
classes dual to various multisingularity loci via so called residue
polynomials associated with the multisingularities. This formula
refines the known multiple point formulas due to
S.L.~Kleiman~\cite{Kl1,Kl2,Kl3} and S.~Katz~\cite{Katz} to the case
of arbitrary proper maps and generalizes it to the case of arbitrary
multisingularity types.

The formula reduces the problem of finding classes of
multisingularity loci to the problem of finding the coefficients of
the residue polynomial. To find these coefficients we use a method of
using symmetries suggested by R.~Rim\'anyi for a close
problem~\cite{R1,R2}. Note that these computations are very far from
the concrete geometrical problems to which the formulas are applied.
This is the reason, perhaps, why our main formula have not been
observed before.

We present also a version of this formula for the classes dual to the
loci of isolated hypersurface multisingularities. Particular cases of
this formula cover many classical enumerative problems: enumerating
singular curves on surfaces, enumerating singular hyperplane sections
of a fixed projective variety, enumerating projective subspaces of a
given tangency type with a fixed hypersurface and many others. Our
computations agree with the classical formulas of Pl\"ucker and
Salmon as well as with more recent results of P.~Le Barz~\cite{LB},
S.J.~Colley~\cite{Co1,Co2}, V.A.~Vainsencher~\cite{Vains},
L.~G\"ottsche~\cite{G}, L.~Caporaso and J.~Harris~\cite{CH},
S.L.~Kleiman and R.~Piene~\cite{KP1,KP2} and many other authors
contributed to the solution of these problems. In particular, our
formula confirms and refines the G\"ottsche's conjecture on the
number of plane nodal curves and all its variations due to Kleiman
and Piene. 

We recognize that the present paper contains no proof of the main
formula. Therefore until the complete proof is published it should
be considered as a conjecture. But after numerous topological and
enumerative confirmations there is no doubt that it is valid also
for Chow rings and over any algebraically closed ground field.


\section{Multisingularities of maps}

\subsection{Thom polynomials}

The {\em local algebra\/} of a map germ $f:(\C^m,0)\to(\C^{m+k},0)$,
$k\ge0$, is the quotient algebra $Q_f=\cO_{\C^m,0}/f^*\fm_{\C^{m+k},0}$ of
the ring $\cO_{\C^m,0}$ of function germs at the origin in the source
space over the ideal generated by the components $f_i$,
$i=1,\dots,m+k$ of the map $f$. We will always assume that $\dim
Q_f<\infty$. Map germs not satisfying this condition form a
space of codimension growing to infinity in the jet space  with the
growth of the order of jets. If the order $K$ of jets is fixed and
sufficiently large then the $K$-jets of map germs with
local algebras isomorphic to the fixed algebra $Q$ form a smooth
(non-closed) submanifold in the jet space. The codimension of this
submanifold (as well as the differential type of the partition  of
the jet space into the strata corresponding to the different
algebras) does not depend on $m$, $K$ ($K\gg0$) but essentially
depends on $k$. This allows to compare map germs acting on the
spaces of different dimensions.

\ex Definition. Two map germs $f_i:(\C^{m_i},0)\to(\C^{m_i+k},0)$,
$i=1,2$, are called {\em stably $\cK$-equivalent\/} if their local
algebras are isomorphic. Thus, for different $k$ we obtain
independent classifications of map germs up to a stable
$\cK$-equivalence. Below by a {\em singularity class\/} we mean a
class of map germs with prescribed local algebra (or with a local
algebra from a given family of algebras; in this case we assume that
the $K$-jets of functions from this class form a smooth semialgebraic
subset in the jet space for sufficiently large $K$). 

Let now $f:M\to N$ be a holomorphic map of nonsingular complex
manifolds such that $k=\dim N-\dim M\ge0$. The points of the source
manifold $M$ are classified according to the local singularities of
the map $f$. For a given $\cK$-singularity class $\a$ denote by
$M(\a)\subset M$ the locus where the germ of $f$ has the given
singularity. By the general principle due to R.~Thom,

{\em the cohomology class Poincar\'e dual to the closure of the locus
$M(\a)$ can be expressed as a universal polynomial \(so called {\em
Thom polynomial}\) in variables $c_1,c_2,\dots$ evaluated on the
relative Chern classes $c_i(f)=c_i(f^*TN-TM)$}.

(To justify this assertion we must assume that the map satisfies some
transversality condition, namely, the jet extension of the map $f$
should be transversal to the closure of the singularity locus $\a$ in
the corresponding jet bundle. Or we can assume that the locus
$M(\a)\subset M$ has the `expected' codimension. In the latter case
the irreducible components of $M(\a)$ should be taken with some
multiplicities prescribed by the scheme structure of these
components. In the general case we claim only that the dual of the
characteristic class given by the Thom polynomial can be represented
by a closed cycle supported on the closure of $M(\a)$. Similar remark
is applied to all classes dual to different singularity loci
considered in this note.)

In this note we present similar {\em a priori\/} description of
cohomology classes dual to the multisingularity loci\footnote{Proofs
will appear elsewhere}.




\subsection{Multisingularity loci $M(\protect\ua),N(\protect\ua)$} 

Assume that $f:M\to N$ is a {\em proper\/} holomorphic map of
manifolds of {\em non-negative relative dimension  $k=\dim N-\dim
M\ge0$}.

By a {\em multisingularity\/} $\ua=(\a_1,\dots,\a_r)$ we mean any 
collection of individual singularities.  With each multisingularity
type $\ua=(\a_1,\dots,\a_r)$ we associate the locus $N(\ua)\subset N$
consisting of points $y\in N$ having $r$ preimages in $M$ with
prescribed singularities of $f$ at these points. Denote by $M(\ua)$
the component of $f^{-1}(N(\ua))\subset M$ corresponding to the
points with $\a_1$-singularity. The locus $N(\ua)$  (resp., $M(\ua)$)
does not depend on the order of the singularities $\a_1,\dots,\a_n$
(resp., $\a_2,\dots,\a_r$). Our goal is to compute the cohomology
classes $\om_{\ua}\subset H^*(M)$ and $\on_{\ua}\in H^*(N)$ dual to the
closures of the loci $M(\ua)$ and $N(\ua)$ respectively considered with
their reduced structures.

We shall often consider the classes $\om_{\ua}$, $\on_{\ua}$ with their
natural multiplicities, namely, we set
 $$m_{\ua}=\#\Aut(\a_2.\dots,\a_r)\;\om_{\ua},\quad
   n_{\ua}=\#\Aut(\a_1.\dots,\a_r)\;\on_{\ua},$$
where $\#\Aut(\ua)=\#\Aut(\a_1,\dots,\a_r)$ denote the number of
permutations $\s\in S(l)$ such that $\a_{\s(i)}=\a_i$ for
$i=1,\dots,r$. If $\ua$ contains $k_j$ singularities of type $\b_j$,
$j=1,2,\dots$ with pairwise different $\b_j$, then
$\#\Aut(\ua)=k_1!\cdot k_2!\cdot\dots$.


Note that the natural
projection $f:M(\ua)\to N(\ua)$ is a covering; the degree of this
covering is equal to the number of  $\a_1$-singularities in the
collection $\ua=(\a_1,\dots,\a_r)$. Therefore, we have an evident
relation
  $$n_{\ua}=f_*m_{\ua}\in H^*(N),$$
where $f_*$ is the Gysin homomorphism (which is defined since $f$
is proper).



\subsection{Chern classes and Landweber-Novikov classes}

Our main result is the expression of the classes $\om_\ua$, $\on_\ua$ in
terms of certain characteristic classes associated with the map $f$.
Those include the Chern classes of the manifolds, $c_i(TM)\in
H^*(M)$, $c_i(TN)\in H^*(TN)$. More classes can be obtained from these
ones using the pull-back homomorphism $f^*:H^*(N)\to H^*(M)$ and the Gysin
or push-forward homomorphism $f_*:H^*(M)\to H^*(N)$ (which is defined
if $f$ is proper). In fact, the final expressions include only
particular combinations of these classes defined as follows.

On the {\em source\/} manifold $M$ we define the {\em relative Chern
classes} $c_i(f)=c_i(f^*TN-TM)$ of $f$. With any formal monomial
$c^I=c_1^{i_1}c_2^{i_2}\dots$ in variables $c_1,c_2,\dots$ we
associate an independent variable $s_I$. Every such variable $s_I$ defines
the cohomology class on the {\em target\/} manifold $N$ (called the {\em
Landweber-Novikov class}):
$$s_I(f)=f_*(c_1(f)^{i_1}c_2(f)^{i_2}\dots)\in H^{2\deg s_I}(N)$$
with $\deg s_I=k+\deg c^I=k+i_1+2i_2+\dots$, where $k=\dim N-\dim M$.

We consider the cohomology algebra $H^*(M)$ as an $H^*(N)$-module via
$f^*$. Having this in mind we often use the same notation for a
class  $\l$ in $H^*(N)$ and for its inverse image $f^*\l$ in
$H^*(M)$. In particular, we preserve the notation $s_I(f)$ for the
corresponding Landweber-Novikov classes
$f^*s_I(f)=f^*\!f_*(c^I(f))\in H^*(M)$ on the source manifold.

More formal, consider the ring $\cC=\Z[c_1,c_2,\dots]$ of polynomials
in abstract Chern classes and define a new polynomial ring
$$\cSC=\Sym\cC=1\oplus\cC\oplus\Sym^2\cC\oplus\dots.$$
of polynomials in abstract Landweber-Novikov classes. The
natural linear map $\cC\to\cSC$
sending a polynomial in Chern classes to the corresponding linear
combination of the Landweber-Novikov classes extends to the
homomorphism of $\cSC$-modules
  $$s:\cC\otimes\cSC\to\cSC.$$
The rings $\cC$, $\cSC$, $\cC\otimes\cSC$ are graded by the agreement 
$\deg c_i=i$ and by the condition that $s$ increases the grading by $k$. 

For any proper map $f:M\to N$ we can define the characteristic
homomorphisms
$$\cC\to H^*(M),\quad \cSC\to H^*(N),\quad \cC\otimes\cSC\to H^*(M),$$
evaluating the corresponding characteristic classes associated with
the map $f$. These characteristic homomorphisms make the following
diagram commutative
$$\begin{array}{ccccccc}
\cC\otimes\cSC&\too[s]&\cSC&\too[1\otimes\Id]&\cC\otimes\cSC\\
\downarrow&&\downarrow&&\downarrow\\
H^*(M)&\too[f_*]&H^*(N)&\too[f^*]&H^*(M)
\end{array}$$




\subsection{Main theorem}\label{mainth}

Below in Sections~\ref{cobord}, \ref{res} we define, for any collection
$\ua=(\a_1,\dots,\a_r)$ of monosingularities, the polynomial
$R_{\ua}\in \cC$ in variables $c_1,c_2,\dots$. This polynomial called
the {\em residue polynomial\/} of the multisingularity $\ua$ has the
following properties:

--- the degree of $R_{\ua}$ is $k(r-1)+\sum\codim\a_i$ (with respect to
the grading with $\deg c_i=i$); the (complex) codimension of the
singularity $\a_i$ is counted in the jet space of map germs
$(\C^m,0)\to(\C^{m+k},0)$ that is in the source manifold $M$.

--- the polynomial $R_{\ua}$ does not depend on the order of the
singularities $\a_i$ in the collection $(\a_1,\dots,\a_r)$.

--- if the collection $\ua=\{\a\}$ consists of the only singularity class
($r=1$) then $R_\a$ is the corresponding Thom polynomial.

\medskip
Now, denote by $S_{\ua}=s(R_{\ua})\in \cSC$ the corresponding linear
combination of the Landweber-Novikov classes.

If $f:M\to N$ is a holomorphic map of complex manifolds then, as
explained above, $R_{\ua}$ and $S_{\ua}$ define the corresponding
characteristic cohomology classes on $M$ and $N$. 


\th Theorem 1. {For each $k\ge0$ and any collection
$\ua=(\a_1,\dots,\a_r)$ of local singularity classes there exists
a universal polynomial $n_\ua\in \cSC$ \(resp., $m_\ua\in
\cC\otimes\cSC$\) in Landweber-Novikov classes $s_I$ \(resp., Chern
classes and Landweber-Novikov classes $c_i,s_I$\) such that for any
proper holomorphic map $f:M\to N$ of manifolds of relative dimension
$k=\dim N-\dim M$ the cohomology class dual to the corresponding
multisingularity locus of the map $f$ is the corresponding
characteristic class evaluated on the given map $f$.

The polynomials $m_\ua$, $n_\ua$ are defined
completely by the residue polynomials
$R_{\ua_J}=R_{\a_{j_1},\dots,\a_{j_l}}$ for various subsets
$J=\{j_1,\dots,j_l\}\subset\{1,\dots,r\}$. Explicitly, the following
relations hold
\begin{eqnarray}
m_{\ua}&=&
\!\!\!\sum_{\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l}\!\!\! 
R_{\ua_{J_1}}\;S_{\ua_{J_2}}\dots S_{\ua_{J_l}},\label{eq1}\\
n_{\ua}&=&s(m_{\ua})=
\!\!\!\sum_{\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l}\!\!\! 
S_{\ua_{J_1}}\dots S_{\ua_{J_l}}.
\label{eq2}\end{eqnarray}
The summation is taken over all possible partitions of the set
$\{1,\dots,r\}$ into a disjoint union of nonempty subsets
$\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l$, $l\ge1$. The subset
containing the first element $1\in\{1,\dots,r\}$ is denoted by $J_1$.}

\ex Remark. The residue polynomials $R_{\ua}$ are defined over
integers and the relations~(1--2) hold also over integers. On the
other hand, the expressions for the classes 
$\om_{\ua}=m_{\ua}/\#\Aut(\a_2.\dots,\a_r)$ and 
$\on_{\ua}=n_{\ua}/\#\Aut(\ua)$ of reduced multisingularity loci in
terms of the Chern-Landweber-Novikov classes may contain rationals
and are valid only over $\Q$.

Combining terms in the right hand side expressions we arrive
at the following recursive formulas which are equivalent to~(1--2).
\begin{eqnarray}
m_{\ua}&=&
\!\!\!\sum_{1\in J\subset\{1,\dots,r\}}\!\!\! 
R_{\ua_{J}}\;n_{\ua_{\overline J}},\label{eq3}\\
n_{\ua}&=&
\!\!\!\sum_{1\in J\subset\{1,\dots,r\}}\!\!\! 
S_{\ua_{J}}\;n_{\ua_{\overline J}},
\label{eq4}\end{eqnarray}
where the summation is taken over all subsets $J\subset\{1,\dots,r\}$
containing the first element $1$, and $\overline J=\{1,\dots,r\}\setminus J$.

For example,
$$\begin{array}{rcl}
m_{\a}&=&R_{\a},\\
n_{\a}=s(m_{\a})&=&S_{\a},\\
m_{\a,\b}&=&R_{\a,\b}+R_{\a}\;S_{\b},\\
n_{\a,\b}=s(m_{\a,\b})&=&
S_{\a,\b}+S_{\a}\;S_{\b},\\
m_{\a,\b,\g}&=&R_{\a,\b,\g}+
R_{\a,\g}\;S_{\b}+R_{\a,\b}\;S_{\g}%\\&&~
   +R_{\a}\;(S_{\b,\g}+S_{\b}\;S_{\g}),\\
n_{\a,\b,\g}=s(m_{\a,\b,\g})&=&
S_{\a,\b,\g}+S_{\a,\b}\;S_{\g}+
    S_{\b,\g}\;S_{\a}+S_{\a,\g}\;S_{\b}+S_{\a}\;S_{\b}\;S_{\g},\\
&&\quad\hbox{etc}.
\end{array}$$


Note that~(\ref{eq1}) or~(\ref{eq3}) define $R_{\ua}$ recursively
via $m_{\ua}$ and $n_{\ua}=s(m_{\ua})$ (see Sect.~\ref{cobord}):
\begin{eqnarray*}
R_{\a}&=&m_{\a},\\
R_{\a,\b}&=&m_{\a,\b}-
m_{\a}n_{\b},\\
R_{\a,\b,\g}&=&m_{\a,\b,\g}+
2m_{\a}\;n_{\b}\;n_{\g}%\\&&~
-m_{\a}\;n_{\b,\g}-
m_{\a,\b}\;n_{\g}-
m_{\a,\b}\;n_{\g},\\
&&\quad\dots
\end{eqnarray*}
\begin{eqnarray}
R_{\ua}&=&\displaystyle
\!\!\!\sum_{\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l}\!\!\! 
(-1)^{l-1} (l-1)!\;
m_{\ua_{J_1}}\;n_{\ua_{J_2}}\dots n_{\ua_{J_l}},\label{inverse}
\end{eqnarray}

In this form Theorem~1 asserts that for any proper map $f:M\to N$ 
the cohomology class on the source manifold $M$ given
by the right hand side combination~(\ref{inverse})
of the classes $m_{\ua}=\#\Aut(\a_2,\dots,\a_r)[M(\ua)]$ and
$n_{\ua}=f^*f_*m_{\ua}$ can be expressed as
a universal polynomial in relative Chern classes $c_i(f)$.

\ex Example. Denote by $A_r$ the singularity class with the local
algebra isomorphic to $\C[x]/\{x^{r+1}\}$ (i.e.\ the singularity class
of Boardman type $\S^{1,\dots,1}$, $r$ entries). Then, for $k=1$, we
have
  $$R_{A_0}=\om_{A_0}=1;\qquad R_{A_1}=\om_{A_1}=c_2,\qquad
R_{A_0,A_1}=-2c_1c_2-2c_3.$$
The cohomology classes dual to different multisingularity loci
corresponding to the classes $A_0,A_1$ are
\begin{equation}\begin{array}{r@{}l}
\om_{A_0,A_1}&{}=
f^*\!f_*(c_2)-2c_1c_2-2c_3=s_{0,1}-2c_1c_2-2c_3,\\
\om_{A_1,A_0}&{}=
c_2\;f^*\!f_*(1)-2c_1c_2-2c_3=c_2s_{0}-2c_1c_2-2c_3,\\
\on_{A_0,A_1}&{}=f_*\om_{A_0,A_1}=f_*\om_{A_1,A_0}\\
&{}=f_*(1)f_*(c_2)-2f_*(c_1c_2+c_3)\\
&{}=s_0s_{0,1}-2s_{1,1}-2s_{0,0,1}.
\end{array}\label{eq5}\end{equation}



\subsection{Example: multiple point formulas}\label{exmul}

Assume that the multisingularity class $\ua=\a^r=(\a,\dots,\a)$ is
formed by $r$ monosingularities of the same type $\a$. Then by 
Theorem~1, the corresponding cohomology classes $\om_{\a^r}$,
$\on_{\a^r}$  can be expressed in terms of the polynomials
$q_p=q_p(\a)=\frac1{p!}R_{\a^{p+1}}=\frac1{p!}R_{\a,\dots,\a}$ ($p+1$
entries) of degree $(k+1)\codim\a+k\,r$. For instance, if $k=1$ and
$\a=A_0$ (the class of non-singular map germs) then these polynomials
are
$$q_0=1,\qquad q_1=-c_1,\qquad q_2=c_1^2+c_2,\qquad
  q_3=-c_1^3-3c_1c_2-2c_3,\qquad\dots.$$
For the case $k=0$, $\a=A_1$ (the folding singularity) these
polynomials are
$$q_0=c_1,\qquad q_1=-2(2c_1^2+c_2),
\qquad q_2=4(5c_1^3+7c_1c_2+3c_3),\qquad\dots.$$

If we set $v_i=v_i(\a)=\frac1if_*(q_{i-1})$ and write the formal series 
$$\begin{array}{ll}
m(t)=\om_1+\om_2t+\om_3t^2+\dots\in\cC\otimes\cSC\otimes\Q[[t]],\\
q(t)=q_0+q_1t+q_2t^2+\dots\in\cC\otimes\Q[[t]],\\ 
n(t)=1+\on_1t+\on_2t^2+\dots\in\cSC\otimes\Q[[t]],\\
v(t)=v_1t+v_2t^2+v_3t^3+\dots\in\cSC\otimes\Q[[t]]
\end{array}$$
then relations~(\ref{eq2}) and~(\ref{eq4}) in this case take the form of the following
two equations equivalent to each other
  $$n(t)=\exp(v(t)),\quad\hbox{and}\quad 
    n'(t)=v'(t)\;n(t)\quad(~{}'={d}/{dt}),$$
that is
$$\on_1=v_1,\quad 2 \on_2=2v_2+v_1^2,\quad
6\on_3=6v_3+6v_1v_2+v_1^3,\quad\dots.$$
(these polynomial expressions of $\on_i$ in terms of $v_i$ are known,
up to a rescaling, as {\em Bell polynomials}). Expressions for $\om_i$ are
obtained in a similar way. From~(\ref{eq3}) one gets the relation
$m(t)=q(t)\;n(t)$ which can be written as the following recursive
{\em formula for the classes of multiple singularity $\a$}:
$$\om_{r}=\sum_{i=1}^{r}q_i\,\on_{r-i},
\qquad \on_{r}=r\,f_*(\om_{r}).$$

Note that if $\a=A_0$ is the class of non-singular maps then
$q_0=q_0(A_0)=1$ and the last relation can be rewritten in the form
  $$n(t)=p(t)\;m(t),\quad
p(t)=1+p_1t+p_2t^2+\dots=q^{-1}(t)\in\cC\otimes\Q[[t]].$$
or
  \begin{equation}
\om_{r}=f^*(\on_{r-1})-\sum_{i=1}^{r-1}p_{i}\;\om_{r-i}.
\label{eq6}\end{equation}

The existence of the polynomials $p_r$ in relative Chern classes
$c_i(f^*TN-TM)$ satisfying relations~(\ref{eq6}) has been proved by
S.L.~Kleiman~\cite{Kl2,Kl3} and S.~Katz~\cite{Katz} under the
assumption that $f$ is a Morin map that is it has only singularities
of corank $1$. Rim\'anyi~\cite{R2} has shown that with a better
choice of $p_6$, $p_7$ (for $k=1$) the formula~(\ref{eq6}) holds for
$r\le 8$ for all stable maps. We claim that~(\ref{eq6}) {\em holds
for any proper map and for all $r\ge1$. Moreover, the polynomials
$p_r$ are uniquely defined by this condition}.

\subsection{Operations in cobordism theories}\label{cobord}


In this section we explain the relation between the theory of
characteristic classes of multisingularities and the cobordism
theory. These results serve as the motivation for the main theorem of
Sect.~(\ref{mainth}). They may also be considered as the explicit
description of the `generalized Pontryagin-Thom construction' of
Sz\"ucs and Rim\'anyi~\cite{SR}.

\subsubsection{Real case.} Consider a smooth proper map $f:M\to
N$ of smooth {\em real\/} manifolds, $k=\dim_\R N-\dim_\R M\ge0$. The
closure of a multisingularity locus $N(\ua)\subset N$ is a locally
semialgebraic set an its dual cohomology class is not always defined
(even in the cohomology {\em modulo}~$2$). Such cohomology classes can be
defined only for certain linear combinations of multisingularities
satisfying some boundary conditions (see~\cite{Vassiliev}). Let
$\chi(f)\in H^*(N,\Z_2)$ be a class defined in this way.

The map $f$ can be considered as a representative of a
$k$-dimensional cobordism class on $N$. It is easy to observe that
the class $\chi$ is an invariant of the cobordism class, in
other words, if the maps $f:M\to N$ and $f':M'\to N$ are cobordant
then $\chi(f)=\chi(f')$. Moreover, we have

\th Theorem 2. Whenever the $d$-dimensional cohomology class \(modulo
$2$, for definiteness\) dual to some linear  combination of
codimension $d$ multisingularity loci is correctly defined for any
generic proper map $f:M\to N$ with $\dim N-\dim M=k$, it provides a
universal cohomology operation from the $k$-dimensional cobordisms
$O^k(\cdot)$ to the $d$-dimensional cohomology $H^d(\cdot,\Z_2)$.

The classification of all possible operations can be obtained from
the classical Thom construction. A possible way to describe this
construction is as follows: there exists a `universal $k$-dimensional
cobordism' ${\bf f}:{\bf M}\to{\bf N}$ such that any cobordism class
on any manifold $N$ is induced from ${\bf f}$ by some continuous map
$N\to{\bf N}$ defined uniquely up to homotopy. For the space ${\bf
N}$ we can take the iterated loop space $\O^mMO(m+k)$, $m\gg0$, where
$MO(n)$ is the Thom space of the classifying $n$-dimensional vector
bundle over $BO(n)$. Therefore, by usual arguments, we obtain the
following general result.

\th Proposition. The cohomology operations $O^k(\cdot)\to
H^d(\cdot,\Z_2)$ are in one-to-one correspondence with the elements
of the cohomology group $H^d(\O^mMO(m+k),\Z_2)$, $m\gg0$ \(these
groups stabilize for sufficiently large $m$\).





\subsubsection{Complex case.} A proper holomorphic map $f:M\to N$ of
complex manifolds can be treated as a representative of a {\em
unitary} $2k$-dimensional cobordism class $[f]\in U^{2k}(N)$ with
$k=\dim_\C M-\dim_\C N$. In general, a unitary cobordism class is not
necessary represented by a holomorphic map. Nevertheless, we may
expect that the correspondence associating with a map $f:M\to N$ the
cohomology class on $N$ dual to the closure of some multisingularity
locus extends similarly to the real case to a universal cohomology
operation $U^{2k}(\cdot)\to H^*(\cdot)$. Theorem~3 below shows that
this is true.

By the same arguments as above the cohomology operations
$U^{2k}(\cdot)\to H^*(\cdot)$ are in one-to-one correspondence with
the elements of the cohomology ring $H^*(\O^{2m}MU(m+k))$,
$m\to\infty$. Fortunately, this ring can be computed explicitly using
Vassiliev's version of Anderson's spectral sequence at least for the
case of rational coefficients, which gives the complete description
of the cohomology operations $U^{2k}(\cdot)\to H^*(\cdot,\Q)$ for
$k>0$.

For every monomial $c^I=c_1^{i_1}c_2^{i_2}\dots$ the corresponding
Landweber-Novikov class $s_I$ is defined for any unitary cobordism
class. The Landweber-Novikov classes are well known in cobordism
theory. In fact, they can be defined in cobordisms but we consider
here only their images in cohomology.

\th Theorem 3. For any $k>0$ there is a one-to-one correspondence
between the operations $U^{2k}(\cdot)\to H^*(\cdot,\Q)$ and the elements
of the polynomial ring in the Landweber-Novikov classes $s_I$. 

Due to this theorem the first assertion of Theorem~1 concerning the
multisingularity classes in the target manifold can be
reformulated as follows.

\th Proposition. For any multisingularity class $\ua=(\a_1,\dots,\a_r)$
the correspondence $f\mapsto \on_\ua\in H^*(N)$ defined on proper holomorphic
maps $f:M\to N$ extends to a universal cohomology operation
$U^{2k}(\cdot)\to H^*(\cdot,\Q)$, and hence, $\on_\ua$ can be expressed
as a universal polynomial with rational coefficients in the
Landweber-Novikov classes.

Let us show that the explicit formula~(\ref{eq2}) of Theorem~1 follows
from this proposition. Indeed, {\em define\/} the classes $S_\ua\in H^*(N)$
for $\ua=(\a_1,\dots,\a_r)$ by setting $S_\a=n_\a$ if
$\ua=\{\a\}$, $r=1$, and for $r>1$ inductively by the relation
\begin{equation}
n_\ua=S_{\ua}+
\!\!\!\sum_{\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l\atop l\ge2}\!\!\! 
S_{\ua_{J_1}}\dots S_{\ua_{J_l}}.
\label{induct}
\end{equation}

By the proposition above the classes $S_\a$ are expressed as certain
polynomials in Landweber-Novikov classes. We would like to show that
these polynomials are of degree one that is {\em linear\/} in
Landweber-Novikov classes. 

Since these polynomials are universal the relation~(\ref{induct})
holds for any map. We wold like to apply it to the sequence of maps
$f^{(d)}:M^{(d)}\to N$, $d=1,2,\dots$ defined as follows. Assume that
the map $f:M\to N$ is included into a family of maps from $M$ to $N$
such that the generic members of this family are `sufficiently
generic' with respect to each other. Define
$M^{(d)}=M\sqcup\dots\sqcup M$ to be the disjoint union of $d$ copies
of $M$ and the restriction $f_i:M\to N$ of $f^{(d)}$ to the $i$th
component of $M^{(d)}$ to be a small generic perturbation of $f$.

Since each $f_i$ is homotopy equivalent to $f$ we have that the
classes $n_\ua=n_\ua(f_i)=n_\ua(f)$ are independent on $i$. For the
map $f^{(d)}$ the $r$ preimages of a point from the locus of the
multisingularity $\ua=(\a_1,\dots,\a_r)$ can be distributed arbitrary
between the components of $M^{(d)}$. Therefore, for the class
$n^{(d)}_\ua$ corresponding to this locus we have an evident relation
\begin{eqnarray}  
n^{(d)}_\ua&=&\!\!\!\sum_{\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l}\!\!\! 
d(d-1)...(d-l+1)\;n_{\ua_{J_1}}\dots n_{\ua_{J_l}}\nonumber\\
&=&\sum_{l=1}^r d(d-1)...(d-l+1)\;n_\ua(l),\label{eq9}
\end{eqnarray}
where by $n_\ua(l)$ we denote the sum of the products of the form
$n_{\ua_{J_1}}\dots n_{\ua_{J_l}}$ with the fixed number $l$ of
subsets in the partition $\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l$.
Now, it follows from~(\ref{induct}) that $n_\ua(l)$ is expressed via
the classes $S_{\ua_J}$ for various subsets $J\subset\{1,\dots,r\}$ as
$$  
n_\ua=\!\!\!\sum_{\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_p}\!\!\! 
c_{p,l}\;S_{\ua_{J_1}}\dots S_{\ua_{J_p}}
=\sum_{p=l}^r c_{p,l}\;S_\ua(p),
$$
where $S_\ua(p)$ is the sum of the products of the form
$S_{\ua_{J_1}}\dots S_{\ua_{J_p}}$ with the fixed number $p$ of
subsets in the partition $\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_p$ and
the constant $c_{p,l}$ is the number of all possible partitions of a
$p$-element set into a disjoint union of $l$ non-empty non-ordered
subsets (these numbers are called the {\em Stirling numbers of the second
kind\/} and denoted usually by $S(p,l)$, cf.~\cite{Cameron}, but we use
the notation $c_{p,l}$ in order do not confuse with the classes
$S_\ua$). Substituting this to~(\ref{eq9}) we get
$$n^{(d)}_\ua=\sum_{1\le l\le p\le r} 
         d(d-1)...(d-l+1)\;c_{p,l}\;S_\ua(p).$$
Using the well-known identity~\cite{Cameron}
 $\sum_{l=1}^p d(d{-}1)...(d{-}l{+}1)\;c_{p,l}=d^p$
we get finally
\begin{equation}
n^{(d)}_\ua=\sum_{p=1}^r d^p\;S_\ua(p)=
\!\!\!\sum_{\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_p}\!\!\! 
d^p\;S_{\ua_{J_1}}\dots S_{\ua_{J_p}}.
\label{eq10}
\end{equation}

Now compute the Landweber-Novikov classes of the map $f^{(d)}$. It is
clear that the relative Chern classes of $f^{(d)}$ coincide with
those of $f$ on each component of $M^{(d)}$. Hence
$s_I(f^{(d)})=d\,s_I(f)$ that is passing from $f$ to $f^{(d)}$ acts
on the polynomial $n_\ua$ as the homogeneous stretching of the
variables $s_I$. Therefore the linear terms of $n_\ua$ coincide with the
coefficient of the linear term in $d$ in the expression for
$n_\ua^{(d)}$. The formula~(\ref{eq10}) show that this term is equal
to $S_\ua$. This complete the proof of the fact that $S_\ua$ is a linear
combination Landweber-Novikov classes and is given by $S_\ua=s(R_\ua)$
for certain polynomial $R_\ua$ in relative Chern classes.

\mdskip
Note that extracting the linear term in $d$ from the
relation~(\ref{eq9}) we get the following explicit
formula for $S_\ua$ which inverts the relation~(\ref{induct}),
 $$S_\ua=%\sum_{l=1}^r (-1)^{l-1}\,(l-1)!\;n_\ua(l)=
\!\!\!\sum_{\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l}\!\!\! 
(-1)^{l-1}\,(l-1)!\;n_{\ua_{J_1}}\dots n_{\ua_{J_l}}.$$

\mdskip A little modification of the arguments above explains more
delicate relations~(\ref{eq1}) and~(\ref{inverse}) of
Sect.~(\ref{mainth}).


\ex Remark. {Since the relation between complex geometry and unitary
cobordism theory is not so straightforward the arguments of this
section cannot be considered as a rigorous proof of Theorem~1 but
rather as a very strong confidence of its validity. There is no
doubts that this theorem holds true for Chow rings instead of
cohomology and for any algebraically closed ground field but the
complete proof (and even the correct definition of the localized
classes $m_\ua$, $n_\ua$ valid for non-generic maps) requires a
substantial background in intersection theory. This will be done
elsewhere.

In fact, the arguments of this section were the starting point of our
investigations. Assuming the existence of the polynomials of the
proposition above we computed them explicitly for a number of
multisingularities using a version of the method suggested by
R.~Rim\'anyi. Then we observed many striking relations between these
polynomials. For example, the (homogeneous) degree of the polynomial
equals the number of monosingularities forming the multisingularity,
the higher homogeneous component is equal, up to a factor
$\#\Aut(\ua)$, to the product of the classes dual to the
corresponding monosingularities, etc. Studying these relations we
obtained finally formulae~(\ref{eq1})--(\ref{eq4}).}




\subsection{Determination of the residue polynomials
$R_{\protect\ua}$} \label{res}

In this section we present a constructive algorithm for computing the
residue polynomials. The simplest way to compute $R_\ua$ for a
particular multisingularity $\ua$ is to use the method invented by
R.~Rim\'anyi for a close problem~\cite{R1,R2}. In this method we rely
on the existence of the polynomial $R_\ua$ provided by Theorem~1.
Namely, we define $R_\ua$ inductively as the unique polynomial in
$c_i$ satisfying the relation
$$R_\ua=m_{\ua}-
\!\!\!\sum_{\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l\atop l\ge2}\!\!\! 
R_{\ua_{J_1}}\;S_{\ua_{J_2}}\dots S_{\ua_{J_l}}.$$

To compute the coefficients of this polynomial we shall consider the
number of examples where we can compute both the
Chern-Landweber-Novikov classes and the classes dual to the
multisingularity loci. Every such example provides relations on the
coefficients of the polynomial $R_\ua$. These relations determine the
polynomial.

Consider a Lie group $H$ acting on $\C^m$ and $\C^{n}$, $n=m+k$. Let
$f_0:\C^m\to\C^{n}$ be a proper stable $H$-equivariant holomorphic
map. Then we may set $M$ and $N$ to be the total spaces of the vector
bundles $\C^{n}_H$, $\C^m_H$ over the classifying space $BH$ of the
group $H$ associated with the action of
$H$ on $\C^m$, $\C^{n}$ respectively, and $f:M\to N$ to be the
fibered map coinciding with $f_0$ on each fiber. Then for this
particular example one has:

--- the source and the target manifolds $M$, $N$ are homotopy
equivalent to $BH$ and $H^*(M)=H^*(N)=H^*(BH)$, the ring of
characteristic classes of $H$;

--- the total relative Chern class is $c(f)=c(\C^{n}_H)/c(\C^m_H)$,
where $c(\C^m_H)$ and $c(\C^{n}_H)$ are the total Chern classes of
the representations of $H$ in $\C^m$ and $\C^{n}$ respectively. These
classes are always easy to compute using the splitting principle;

--- the Landweber-Novikov classes are
$$s_I(f)=f_*(c^I(f))=c^I(f)\;f_*(1),$$  where $f_*(1)$ is determined
by the relation $c_{n}(\C^{n}_H)=c_m(\C^m_H)\;f_*(1)$ (this relation
follows from the commutativity of the diagram
 $$\begin{array}{cccccc}
M&\to[f]&N\\
\uparrow&&\uparrow\\
BH&=&BH
\end{array}$$
where the vertical arrows are embeddings of the zero section);

--- if the map $f_0$ has the singularity $\b$ at the origin of the
codimension not greater than the codimension of $M(\ua)$ then
$M(\ua)$ is empty and $\om_\ua=0$, unless $r=1$ and $\ua=\{\b\}$, in
which case $M(\ua)=M(\b)$ is the zero section of the bundle $M\to BH$ and
$\om_\ua=\om_\b=c_m(\C^m_H)$ is the top Chern class of this bundle.

\th Proposition. In the stable range of dimensions, i.e.\ if for a
given $p$ all monosingularities of codimension $\le p$ are simple,
then the relations described above determine completely the residue
polynomials $R_\ua$ for all multisingularity classes of codimension
$\le p$.

For the map $f_0:\C^m\to\C^{n}$ we take the `normal form' of one of
the singularities of codimension $\le p$ and for $H$ we take the
`symmetry group' of this singularity defined as follows,
cf.~\cite{R1}.

Let $Q$ be a local algebra. Its {\em corank\/} is defined as the
minimal number of generators or the minimal number $m$ such that $Q$
can be realized as the local algebra of some germ
$f:\C^m,0\to\C^n,0$. If the number of generators of $Q$ is $m$
then the minimal number of relations determining $Q$ is $m+d$ where
$d$ is independent of $m$ and is called the {\em defect\/} of~$Q$.


\ex Definition. Let $\a=\a(Q)$ be the class of stable
$\cK$-equivalence of maps determined by the given local algebra $Q$.
The {\em symmetry group\/} $G_\a$ of this singularity is the maximal
compact subgroup in the $\cK$-stabilizer of any representative
$f:\C^c,0\to\C^{c+k},0$ of this class with $c$ the corank of~$Q$.

The explicit structure of the symmetry group is described by the
following not so complicated assertion.

\th Proposition.  The symmetry group $G_a$ is isomorphic to
 $$G_\a\cong\Aut Q\times U(k-d),$$
where $\Aut Q$ is the maximal compact subgroup in the automorphism group of
$Q$ and $d$ is the defect of~$Q$.

\ex Remark. {Thus, the method of Rim\'anyi reduces the problem of
finding the residue polynomials to the problem of solving a system of
linear equations on the coefficients of the polynomial. This system
is highly overdetermined. The following observation allows us to get
rid of a large part of redundant relations and to reduce considerably
the computations. Let $\chi$ be a polynomial of degree $p$ in the
Chern-Landweber-Novikov classes and $l$ be an integer.

{\em Assume that the characteristic class $\chi$ vanishes whenever the map
$f$ possesses only \(multi\)singularities of codimension $<l$. Then for
any map $f$ the restriction of $\chi$ to a tubular neighborhood $U$ of any
$l$-codimensional \(multi\)singularity locus $M(\ua)$ has the form
$\chi|_U=\chi'\, e$, where $e$ is the top Chern class of the
normal bundle of $M(\ua)$ and $\chi'\in H^{2(d-l)}(U)$.}

If some of the coefficients of $\chi$ are unknown then the relations
defining these coefficients should be applied to the class
$\chi'=\chi/e$ of smaller degree $p-l$.}

\ex Remark. More relations can be obtained by considering the normal
forms of {\em multisingularities}. These extra relations allow us to
compute the classes $m_\ua$, $n_\ua$ directly in terms of the
Landweber-Novikov and Chern classes. In our case it is not necessary
because of the explicit expressions of Theorem~1 for these classes in
terms of the residue polynomials~$R_\ua$.




\ex Example. Let $k=1$, and the `sample map' $f:M\to N$ be the zero
section embedding of $M$ to the total space $N$ of a line bundle with
the first Chern class $t$ over $M$. Then $c_1(f)=t$, $c_i(f)=0$ for
$i>1$. On the other hand $f$ has no singularities other then $A_0$.
Therefore, {\em if the multisingularity type $\ua$ contains at list
one singularity other than $A_0$ then the coefficient of
$c_1^{\codim\ua}$ in the residue polynomial $R_\ua$ vanishes}.

\ex Example. {Consider the Whitney umbrella or the pinch-point singularity
$$y_1=x_1^2,\quad y_2=x_1x_2,\quad y_3=x_2.$$
These formulas can be treated as the map $\C^2\to[f_0]\C^3$ of coordinate
spaces or as the map $\xi\oplus\eta\to[f]\xi^2\oplus\xi\eta\oplus\eta$ of
total spaces of vector bundles $M\to B$ and $N\to B$ of ranks $2$ and
$3$ respectively over some base $B$ where $\xi$ and $\eta$ are some
line bundles over $B$ with the first Chern classes $t_1$ and $t_2$
respectively. Then for this particular example we have
$$1+c_1(f)+c_2(f)+c_3(f)+\dots
=\frac{(1{+}2t_1)(1{+}t_1{+}t_2)(1{+}t_2)}{(1+t_1)(1+t_2)}=
1+(2t_1+t_2)+t_1t_2-t_1^2t_2+\dots,$$
and $f_*(1)=\frac{2t_1\,(t_1+t_2)\,t_2}{t_1\,t_2}=2(t_1+t_2)$.

Considering this example is sufficient for computing the coefficients
$a_i$ of the residue polynomial $R_{A_0A_1}=
a_1\,c_1^3+a_2\,c_1c_2+a_3\,c_3$ in the case $k=1$ from the known
polynomials $R_{A_0}=1$, $R_{A_1}=c_2$. Indeed, the locus
$M(A_0A_1)$ of this example is empty and substituting the computed
classes to the equality $0=m_{A_0A_1}=R_{A_0}f_*(R_{A_1})+R_{A_0A_1}$
we obtain the relation
$$8a_1\,t_1^3+(1+12a_1+2a_2-a_3)\,t_1^2t_2
      +(2+6a_1+a_2)\,t_1t_2^2+a_1\,t_2^3=0.$$
Equaling to zero the coefficients we find $a_1=0$, $a_2=a_3=-2$, 
i.e.\ $R_{A_0A_1}=-2c_1c_2-2c_3$. Note that if we take into account
the relation $a_1=0$ of previous example then the last relation
simplifies to $((2+2a_2-a_3)t_1+(2+a_2)t_3)t_1t_2=0$ which agrees with
the assertion of the remark above.}

It is worthwhile to compare the computation above 
with the computation of $R_{A_0A_1}$ (for $k=1$)  provided by
the methods of intersection theory. Denote by $Z\subset M$ the
closure of the singularity locus $M(A_1)$ and set $X=f(Z)\subset N$.
Then we define the class $-R_{A_0A_1}$ as the residual class to
$m_{A_0,A_1}=[M(A_0,A_1)]$ in the intersection product $X\cdot
M=f^*(X)$. To simplify the computations we use the following
observation. {\em If\/ $\Sigma\subset M$ is a subvariety of
codimension $c$ then the inclusion $M\setminus\S\to M$ induces an
isomorphism of all cohomology groups up to degree $2\,(c-1)$}.
Because of that we can always drop without loss of generality all
degeneracies of high codimension and to assume in the computation of
$-R_{A_0A_1}$ that the map $f:M\to N$ has no (multi)singularities of
codimension greater than $3$.

Therefore, we assume that $Z\subset M$ and $X\subset N$ are smooth
embeddings of codimensions $2$ and $3$ respectively. The total
inverse image $f^{-1}(X)$ has two disjoint components,
$f^{-1}(X)=M(A_0,A_1)\sqcup W$, where $W$ corresponds to the points
$x\in f^{-1}(X)$ with $x\in Z$. Then by the principal formula of the
intersection theory the residual class
$-R_{A_0A_1}=f^{*}(X)-M(A_0,A_1)$ is
given by 
 $$-R_{A_0A_1}=\{f^*c(N_XN)\smallfrown s(W,M)\}_{m-3}=
         s(W,M)_{m-3}+f^*c_1(N_XN)\,s(W,M)_{m-2}.$$

It is easy to find $f^*c_1(N_XN)=c_1(f)+c_1(\nu)$ where $\nu=N_ZM$ is
the normal bundle of $Z\subset M$. It remains to compute the Segre
class $s(W,M)$. The locus of $W$ coincides with $Z$ but as a scheme
$W$ is different from $Z$ and even is not reduced. Therefore some
computations are needed. 

Let $\pi:\tM\to M$ be the blow-up along $Z$. The exceptional divisor
$D$ coincides with the projective bundle $D=P(\nu)$ over $Z$. Set
$t=c_1(\cO(1))=-D|_D$. One can verify by the local computations that
the complete inverse image $\tW=\pi^{-1}(W)$ contains the divisor $D$
and the residual scheme $R$ of $D\subset\tW$ can be identified with
$P(\ell)\subset D=P(\nu)$ where $\ell\subset\nu$ is the line bundle
over $Z$ of the kernels of the derivative $df$. Therefore, by the
residue intersection formula~\cite{F},
\begin{equation} 
s(\tW,tM)_{m-1-k}=t^k[D]+\sum_{i=1}^k{k\choose i}t^{k-i}
     s(R,\tM)_{m-1-i}.\label{resint}
\end{equation}
Since $R$ is smooth,
$$s(R,\tM)_{m-1-i}=c_{i-1}(-N_R\tM)\,[R]=
c_{i-1}(-N_RD-N_D\tM|_R)\,c_1(\Hom(\cO(-1),\nu/\ell))\,[D],$$
that is
$$\begin{array}{l}
s(R,\tM)_{m-2}=(t+c_1(\nu/\ell))\,[D],\\
s(R,\tM)_{m-3}=(t+c_1(\nu/\ell))\,(t-c_1(\nu/\ell)-t)[D]=
 -c_1(\nu/\ell)\,(t+c_1(\nu/\ell))\,[D],
\end{array}$$
and so on. Substituting these classes to~(\ref{resint}) we get
$$\begin{array}{rcl}
s(\tW,\tM)_{m-2}&=&(t+2\cdot0+(t+c_1(\nu/\ell)))\,[D]=
  (2t+c_1(\nu/\ell))\,[D]\\
s(\tW,\tM)_{m-3}&=&(t^2+3\cdot0+3t\,(t+c_1(\nu/\ell))-
c_1(\nu/\ell)\,(t+c_1(\nu/\ell)))\,[D]\\&=&
  (4t^2+2t\,c_1(\nu/\ell)-c_1(\nu/\ell)^2)\,[D]
\end{array}$$

Now we apply $\pi_*$. Taking into ackount the relation
$\pi_*t^i[D]=c_{i-1}(-\nu)[Z]$ we get
$$\begin{array}{l}
s(W,M)_{m-2}=\pi_*s(\tW,\tM)_{m-2}=2\,[Z],\\
s(W,M)_{m-3}=\pi_*s(\tW,\tM)_{m-3}=
 (-4c_1(\nu)+2c_1(\nu/\ell))\,[Z]=-2\,(c_1(\nu)+c_1(\l))\,[Z].
\end{array}$$
Substituting these classes we obtain an itermediate answer
  $$-R_{A_0A_1}=s(W,M)_{m-3}+(c_1(f)+c_1(\nu))\,s(W,M)_{m-2}=
   2\,(c_1(f)-c_1(\ell))\,[Z].$$

It remains to compute the classes $[Z]=m_{A_1}$ and $c_1(\ell)\,[Z]$.
Consider the projective bundle $p:P(TM)\to M$ and set
$\t=c_1(\cO(1))$. The differential $df$ gives rise to a section of the
bundle $\Hom(\cO(-1),f^*TN)$ and the manifold $\tZ$ of zeroes of this
section projects isomorphically to $Z$. Moreover, the bundle $\cO(-1)$
over $\tZ$ corresponds to the bundle $\ell$ oover $Z$ under this
isomorphism. Therefore, we have
$$\begin{array}{rcl}
[Z]=p_*([\tZ])&=&p_*(\t^{n+1}+\t^nc_1(f^*TN)+\dots+c_n(f^*TN)\\
&=&c_2(-TM)+c_1(-TM)c_1(f^*TN)+c_2(f^*TN)=c_2(f),\\
-c_1(\ell)\,[Z]=p_*(\t[\tZ])&=
&p_*(\t^{n+2}+\t^{n+1}c_1(f^*TN)+\dots+\t\,c_n(f^*TN)\\
&=&c_3(-TM)+c_2(-TM)c_1(f^*TN)+\cdots+c_3(f^*TN)=c_3(f),
\end{array}$$
and we find finally
 $$-R_{A_0A_1}=2\,(c_1(f)-c_1(\ell))\,[Z]=2\,(c_1(f)c_2(f)+c_3(f)).$$

Note that the fact that $R_{A_0A_1}$ is expressed in terms of the
relative Chern classes $c_i(f)$ (as well as the equality
$R_{A_0A_1}=R_{A_1A_0}$) is obtained with this approach as an
`unexpected' result of long computations.


\subsection{Residue polynomials and multisingularity classes for
$k=0$.}

The main theorem~1 and its corollaries are valid also for the case
$k=0$, i.e. for the case of proper maps $f:M\to N$ of manifolds of
equal dimensions $\dim M=\dim N$. If $N$ is connected then the {\em
degree\/} of $f$ is well defined,
$$d=f_*(1)\in H^0(N)=\Z.$$

Consider some multisingularity type $\ua=(\a_1,\dots,\a_p)$ with
$\codim\a_i>0$. The total inverse image of a generic point $y\in
N(\ua)$ consists of $p$ points $x_1,\dots,x_p$ with singularities
$\a_1,\dots,\a_p$ and another $d-\mu$ points which are nonsingular for
the map $f$ where $\mu=\mu_1+\dots+\mu_p$ and $\mu_i$ is the {\em
local multiplicity\/} of $f$ at $x_i$ defined as the dimension of the
local algebra of the germ $f$ at $x_i$. It follows that the classes
of multisingularities $(\ua,A_0^r)=(\a_1,\dots,\a_p,A_0,\dots,A_0)$
(with $r$ entries $A_0$) are given by
  \begin{eqnarray*}
\om_{\ua,A_0^r}&=&{d-\mu\choose r}\;\om_\ua,\\
m_{\ua,A_0^r}&=&
(d-\mu)(d-\mu-1)\dots(d-\mu-r+1)\;m_\ua.
\end{eqnarray*}

\th Proposition. The residue polynomials $R_{\ua,A_0^r}$ of a
collection of singularities containing classes $A_0$ of local
diffeomorphisms are determined by the residue polynomials $R_\ua$ of
of the remaining singularities,
 \begin{equation}
R_{\ua,A_0^r}=(-1)^{(r-1)}\mu(\mu+1)\dots(\mu+r-1)\;R_\ua,
\label{RA0}
\end{equation}
where $\mu=\mu(\ua)$ is the sum of local multiplicities of the singularities
forming the collection~$\ua$.

The proof is purely combinatorial, it consists of verifying
identity~(\ref{eq1}) of Theorem~1 for the multisingularity classes
$(\a_1,\dots,\a_p,A_0,\dots,A_0)$. Note that the equality~(\ref{RA0})
is valid also if the collection $\ua$ contains $A_0$-singularities and
so it is sufficient to carry on the computations only for the case $r=1$
which is not difficult.

\ex Example. The numbers expressing the `classes of multiple points'
\begin{eqnarray*}
\om_r&=&\om_{A_0^r}={d-1\choose r-1}\in H^0(M)=\Z,\\
\on_r&=&\on_{A_0^r}={d\choose r}\in H^0(N)=\Z
\end{eqnarray*}
can be expressed via the `residue classes'
$q_r=\frac1{r!}R_{A_0^{r+1}}=(-1)^r$ by the formulas of
Section~\ref{exmul}. In particular, we have
 $$p(t)=(1+q_1t+q_2t^2+\dots)^{-1}=1+t$$
and the 'multiple point formula' for the case $k=0$ takes the form
  $$\om_{r+1}=f^*(\on_{r})-\om_r\in H^0(M)=\Z$$
which is equivalent to the evident relation 
${d-1\choose r}={d\choose r}-{d-1\choose r-1}$.

The proposition reduces the problem of computing the residue
polynomials $R_\ua$ to the case when all the singularities of the
collection $\ua=(\a_1,\dots,\a_r)$ have positive codimensions.

Up to codimension $8$ all possible $\cK$-singularities of maps of
equally dimensional manifolds are listed in the table below
(cf.~\cite{Mather}). The table contains also various information
about these singularities needed for computing residue polynomials.

{\renewcommand{\arraystretch}{1.8}
$$\begin{array}{|c|c|c|c|c|c|}\hline
\a&Q_\a&\codim&H^*(BG_\a,\Q)&c&\mu{=}f_*(1)\\\hline\hline
A_p,~p\ge0&\frac{\C[x]}{x^{p+1}}&p&\Q[t]&
\frac{1+(p+1)t}{1+t}&p+1\\\hline
I_{2,2}&\frac{\C[x,y]}{x^2,~y^2}&4&\Q[t,u]\atop\deg u=2&
\frac{1+2t+4u}{1+t+u}&4\\\hline
I_{p,q},~2{\le} p{\le} q,\;p{+}q{>}4&\frac{\C[x,y]}{xy,~x^p+y^q}&
p{+}q&\Q[t]&\frac{(1+(p+q)t)(1+pqt)}{(1+pt)(1+qt)}&
p{+}q\\\hline
  &\frac{\C[x,y]}{x^2,~y^3}&7&\Q[t,t_1]&
\frac{(1+2t)(1+3t_1)}{(1+t)(1+t_1)}&6\\\hline
  &\frac{\C[x,y]}{x^2+y^3,xy^2}&8&\Q[t,t_1]&
\frac{(1+6t)(1+7t_1)}{(1+2t)(1+3t_1)}&7\\\hline
\end{array}$$}



In the tables below we present the lists of polynomials $R_\ua$,
$\om_\ua$, $\on_\ua$ of small degrees for the case $k=0$. In
fact, we have computed $R_\ua$ (and thus, the classes dual to
the multisingularity loci $\om_\ua,\on_\ua$) for all (multi)singularity
classes $\ua$ of codimension $\le8$. The complete list of computed
polynomials occupies many pages and can be found at the author's
homepage~\cite{wwwkaz}. 


{\small 
\subsubsection*{Residue classes $R_\ua$ of codimension $\le4$ for
$k=0$}
\begin{eqnarray*}
R_{A_1} &=& c_1 \\
R_{A_2} &=& c_1^2 + c_2 \\
R_{A_1^2} &=& -2\,( 2\,c_1^2 + c_2 ) \\
R_{A_3} &=& c_1^3 + 3\,c_1\,c_2 + 2\,c_3 \\
R_{A_1 A_2} &=& -6\,( c_1^3 + 2\,c_1\,c_2 + c_3 ) \\
R_{A_1^3} &=& 8\,( 5\,c_1^3 + 7\,c_1\,c_2 + 3\,c_3 ) \\
R_{A_4} &=& c_1^4 + 6\,c_1^2\,c_2 + 2\,c_2^2 + 9\,c_1\,c_3 + 6\,c_4 \\
R_{I_{22}} &=& c_2^2 - c_1\,c_3 \\
R_{A_1 A_3} &=& -4\,( 2\,c_1^4 + 9\,c_1^2\,c_2 + 2\,c_2^2 +
  11\,c_1\,c_3 + 6\,c_4 ) \\
R_{A_2^2} &=& -3\,( 3\,c_1^4 + 12\,c_1^2\,c_2 + 4\,c_2^2 +
  13\,c_1\,c_3 + 8\,c_4 ) \\
R_{A_1^2 A_2} &=& 24\,( 3\,c_1^4 + 10\,c_1^2\,c_2 + 2\,c_2^2 +
  10\,c_1\,c_3 + 5\,c_4 ) \\
R_{A_1^4} &=& -48\,( 14\,c_1^4 + 37\,c_1^2\,c_2 + 6\,c_2^2 + 
  33\,c_1\,c_3 + 15\,c_4 ) 
%\end{array}$$
\end{eqnarray*}


\subsubsection*{Multisingularity classes
$m_{\a_1,\ua'}=\#\Aut(\ua')\,\om_{\a_1,\ua'}$ of codimension $\le4$
for $k=0$}
\begin{eqnarray*}
m_{A_1,A_1} &=& s_1\,c_1 - 4\,c_1^2 - 2\,c_2 \\
m_{A_1,A_2} &=& s_2\,c_1 + s_0 1\,c_1 - 6\,c_1^3 - 12\,c_1\,c_2 -
  6\,c_3 \\
m_{A_2,A_1} &=& s_1\,c_1^2 - 6\,c_1^3 + s_1\,c_2 - 12\,c_1\,c_2 -
  6\,c_3 \\
m_{A_1,A_1^2} &=& s_1^2\,c_1 - 4\,s_2\,c_1 - 2\,s_{01}\,c_1 - 
  8\,s_1\,c_1^2 + 40\,c_1^3 - 4\,s_1\,c_2 + 56\,c_1\,c_2 + 24\,c_3 \\
m_{A_1,A_3} &=& s_3\,c_1 + 3\,s_{11}\,c_1 + 2\,s_{001}\,c_1 - 
  8\,c_1^4 - 36\,c_1^2\,c_2 - 8\,c_2^2 - 44\,c_1\,c_3 - 24\,c_4 \\
m_{A_2,A_2} &=& s_2\,c_1^2 + s_{01}\,c_1^2 - 9\,c_1^4 + s_2\,c_2 +
  s_{01}\,c_2 - 36\,c_1^2\,c_2 - 12\,c_2^2 - 39\,c_1\,c_3 - 24\,c_4 \\
m_{A_3,A_1} &=& s_1\,c_1^3 - 8\,c_1^4 + 3\,s_1\,c_1\,c_2 -
  36\,c_1^2\,c_2 - 8\,c_2^2 + 2\,s_1\,c_3 - 44\,c_1\,c_3 - 24\,c_4 \\
m_{A_1,A_1 A_2} &=& s_1\,s_2\,c_1 - 6\,s_3\,c_1 + 
  s_1\,s_{01}\,c_1 - 12\,s_{11}\,c_1 - 6\,s_{001}\,c_1 - 
  4\,s_2\,c_1^2 - 4\,s_{01}\,c_1^2 \\&&\qquad{}
  - 6\,s_1\,c_1^3 + 72\,c_1^4 -  2\,s_2\,c_2 - 2\,s_{01}\,c_2 -
  12\,s_1\,c_1\,c_2 + 240\,c_1^2\,c_2 + 48\,c_2^2 \\&&\qquad{}
  - 6\,s_1\,c_3 + 240\,c_1\,c_3 + 120\,c_4 \\
m_{A_2,A_1^2} &=& s_1^2\,c_1^2 - 4\,s_2\,c_1^2 - 2\,s_{01}\,c_1^2
  - 12\,s_1\,c_1^3 + 72\,c_1^4 + s_1^2\,c_2 - 4\,s_2\,c_2 -
  2\,s_{01}\,c_2\\&&\qquad{} 
  - 24\,s_1\,c_1\,c_2 + 240\,c_1^2\,c_2 + 48\,c_2^2 -
  12\,s_1\,c_3 + 240\,c_1\,c_3 + 120\,c_4 \\
m_{A_1,A_1^3} &=& s_1^3\,c_1 - 12\,s_1\,s_2\,c_1 + 40\,s_3\,c_1 - 
  6\,s_1\,s_{01}\,c_1 + 56\,s_{11}\,c_1 + 24\,s_{001}\,c_1 - 
  12\,s_1^2\,c_1^2 \\&&\qquad{}
  + 48\,s_2\,c_1^2 + 24\,s_{01}\,c_1^2 + 
  120\,s_1\,c_1^3 - 672\,c_1^4 - 6\,s_1^2\,c_2 + 24\,s_2\,c_2 +
  12\,s_{01}\,c_2  \\&&\qquad{} 
  + 168\,s_1\,c_1\,c_2 - 1776\,c_1^2\,c_2 - 
  288\,c_2^2 + 72\,s_1\,c_3 - 1584\,c_1\,c_3 - 720\,c_4
\end{eqnarray*}

\subsubsection*{Multisingularity classes $n_\ua=\#\Aut(\ua)\,\on_\ua$
of codimension $\le4$ for $k=0$}
\begin{eqnarray*}
n_{A_1} &=& s_1 \\
n_{A_2} &=& s_2 + s_{01} \\
n_{A_1^2} &=& s_1^2 - 4\,s_2 - 2\,s_{01} \\
n_{A_3} &=& s_3 + 3\,s_{11} + 2\,s_{001} \\
n_{A_1 A_2} &=& s_1\,s_2 - 6\,s_3 + s_1\,s_{01} - 12\,s_{11} -
  6\,s_{001} \\
n_{A_1^3} &=& s_1^3 - 12\,s_1\,s_2 + 40\,s_3 - 6\,s_1\,s_{01} + 
  56\,s_{11} + 24\,s_{001} \\
n_{A_4} &=& s_4 + 2\,s_{02} + 6\,s_{21} + 9\,s_{101} +
  6\,s_{0001} \\
n_{I_{22}} &=& s_{02} - s_{101} \\
n_{A_1 A_3} &=& s_1\,s_3 - 8\,s_4 - 8\,s_{02} + 3\,s_1\,s_{11}
  - 36\,s_{21} + 2\,s_1\,s_{001} - 44\,s_{101} -
  24\,s_{0001} \\
n_{A_2^2} &=& s_2^2 - 9\,s_4 + 2\,s_2\,s_{01} + s_{01}^2 -
  12\,s_{02} - 36\,s_{21} - 39\,s_{101} - 24\,s_{0001} \\
n_{A_1^2 A_2} &=& s_1^2\,s_2 - 4\,s_2^2 - 12\,s_1\,s_3 + 72\,s_4
  + s_1^2\,s_{01} - 6\,s_2\,s_{01} - 2\,s_{01}^2 + 48\,s_{02} - 
  24\,s_1\,s_{11} \\&&\qquad{}
  + 240\,s_{21} - 12\,s_1\,s_{001} + 
  240\,s_{101} + 120\,s_{0001} \\
n_{A_1^4} &=& s_1^4 - 24\,s_1^2\,s_2 + 48\,s_2^2 + 160\,s_1\,s_3
  - 672\,s_4 - 12\,s_1^2\,s_{01} + 48\,s_2\,s_{01} + 
  12\,s_{01}^2 \\&&\qquad{}
  - 288\,s_{02} + 224\,s_1\,s_{11} - 1776\,s_{21} + 
  96\,s_1\,s_{001} - 1584\,s_{101} - 720\,s_{0001}
\end{eqnarray*}


}

\subsection{Residue polynomials and multisingularity classes for
$k=1$.}

In the tables below we present the lists of polynomials $R_\ua$,
$\om_\ua$, $\on_\ua$ of small degrees for the case $k=1$. In
fact, we have computed $R_\ua$ (and thus, the classes dual to
the multisingularity loci $\om_\ua,\on_\ua$) for all (multi)singularity
classes $\ua$ of codimension $\le8$ in the source manifold
(respectively, up to codimension $9$ in the target manifold). These
singularities include, in particular, certain singularities of corank
$2$ (starting from codimension $6$), see~\cite{wwwkaz}. 

{\small
\subsubsection*{Residue classes $R_\ua$ of codimension $\le5$ for
$k=1$}
\begin{eqnarray*}
R_{A_0} &=& 1 \\
R_{A_0^2} &=& -c_1 \\
R_{A_1} &=& c_2 \\
R_{A_0^3} &=& 2\,( c_1^2 + c_2 ) \\
R_{A_0 A_1} &=& -2\,( c_1\,c_2 + c_3 ) \\
R_{A_0^4} &=& -6\,( c_1^3 + 3\,c_1\,c_2 + 2\,c_3 ) \\
R_{A_2} &=& c_2^2 + c_1\,c_3 + 2\,c_4 \\
R_{A_0^2 A_1} &=& 2\,( 3\,c_1^2\,c_2 + 2\,c_2^2 + 7\,c_1\,c_3 +
  6\,c_4 ) \\
R_{A_0^5} &=& 24\,( c_1^4 + 6\,c_1^2\,c_2 + 2\,c_2^2 + 9\,c_1\,c_3
  + 6\,c_4 ) \\
R_{A_0 A_2} &=& -3\,( c_1\,c_2^2 + c_1^2\,c_3 + 2\,c_2\,c_3 +
  4\,c_1\,c_4 + 4\,c_5 ) \\
R_{A_1^2} &=& -2\,( 2\,c_1\,c_2^2 + c_1^2\,c_3 + 4\,c_2\,c_3 + 
  5\,c_1\,c_4 + 6\,c_5 ) \\
R_{A_0^3 A_1} &=& -24\,( c_1^3\,c_2 + 2\,c_1\,c_2^2 + 4\,c_1^2\,c_3
  + 3\,c_2\,c_3 + 8\,c_1\,c_4 + 6\,c_5 ) \\
R_{A_0^6} &=& -120\,( c_1^5 + 10\,c_1^3\,c_2 + 10\,c_1\,c_2^2 +
  25\,c_1^2\,c_3 + 12\,c_2\,c_3 + 38\,c_1\,c_4 + 24\,c_5 ) 
\end{eqnarray*}

\subsubsection*{Multisingularity classes
$m_{\a_1,\ua'}=\#\Aut(\ua')\,\om_{\a_1,\ua'}$ of codimension $\le4$
for $k=1$}
\begin{eqnarray*}
m_{A_0,A_0} &=& s_0 - c_1 \\
m_{A_0,A_0^2} &=& s_0^2 - s_1 - 2\,s_0\,c_1 + 2\,c_1^2 + 2\,c_2 \\
m_{A_0,A_1} &=& s_{01} - 2\,c_1\,c_2 - 2\,c_3 \\
m_{A_1,A_0} &=& s_0\,c_2 - 2\,c_1\,c_2 - 2\,c_3 \\
m_{A_0,A_0^3} &=& s_0^3 - 3\,s_0\,s_1 + 2\,s_2 + 2\,s_{01} - 
  3\,s_0^2\,c_1 + 3\,s_1\,c_1 + 6\,s_0\,c_1^2 \\&&\qquad{}
  - 6\,c_1^3 + 6\,s_0\,c_2 - 18\,c_1\,c_2 - 12\,c_3 \\
m_{A_0,A_0 A_1} &=& s_0\,s_{01} - 2\,s_{11} - 2\,s_{001} -
  s_{01}\,c_1 - 2\,s_0\,c_1\,c_2 + 6\,c_1^2\,c_2 + 4\,c_2^2 -
  2\,s_0\,c_3 \\&&\qquad{}+ 14\,c_1\,c_3 + 12\,c_4 \\
m_{A_1,A_0^2} &=& s_0^2\,c_2 - s_1\,c_2 - 4\,s_0\,c_1\,c_2 + 
  6\,c_1^2\,c_2 + 4\,c_2^2 - 4\,s_0\,c_3 + 14\,c_1\,c_3 + 12\,c_4 \\
m_{A_0,A_0^4} &=& s_0^4 - 6\,s_0^2\,s_1 + 3\,s_1^2 + 8\,s_0\,s_2
  - 6\,s_3 + 8\,s_0\,s_{01} - 18\,s_{11} - 12\,s_{001} - 
  4\,s_0^3\,c_1 \\&&\qquad{}
  + 12\,s_0\,s_1\,c_1 - 8\,s_2\,c_1 - 8\,s_{01}\,c_1
  + 12\,s_0^2\,c_1^2 - 12\,s_1\,c_1^2 - 24\,s_0\,c_1^3 + 24\,c_1^4 \\&&\qquad{}
  + 12\,s_0^2\,c_2 - 12\,s_1\,c_2 - 72\,s_0\,c_1\,c_2 +
  144\,c_1^2\,c_2 + 48\,c_2^2 - 48\,s_0\,c_3 \\&&\qquad{}
  + 216\,c_1\,c_3 + 144\,c_4
\end{eqnarray*}

\subsubsection*{Multisingularity classes $n_\ua=\#\Aut(\ua)\,\on_\ua$
of codimension $\le5$ for $k=1$}
\begin{eqnarray*}
n_{A_0} &=& s_0 \\
n_{A_0^2} &=& s_0^2 - s_1 \\
n_{A_1} &=& s_{01} \\
n_{A_0^3} &=& s_0^3 - 3\,s_0\,s_1 + 2\,s_2 + 2\,s_{01} \\
n_{A_0 A_1} &=& s_0\,s_{01} - 2\,s_{11} - 2\,s_{001} \\
n_{A_0^4} &=& s_0^4 - 6\,s_0^2\,s_1 + 3\,s_1^2 + 8\,s_0\,s_2 - 
  6\,s_3 + 8\,s_0\,s_{01} - 18\,s_{11} - 12\,s_{001} \\
n_{A_2} &=& s_{02} + s_{101} + 2\,s_{0001} \\
n_{A_0^2 A_1} &=& s_0^2\,s_{01} - s_1\,s_{01} + 4\,s_{02} - 
  4\,s_0\,s_{11} + 6\,s_{21} - 4\,s_0\,s_{001} + 14\,s_{101} +
  12\,s_{0001} \\
n_{A_0^5} &=& s_0^5 - 10\,s_0^3\,s_1 + 15\,s_0\,s_1^2 + 
  20\,s_0^2\,s_2 - 20\,s_1\,s_2 - 30\,s_0\,s_3 + 24\,s_4 + 
  20\,s_0^2\,s_{01} \\&&\qquad{}
  - 20\,s_1\,s_{01} + 48\,s_{02} - 90\,s_0\,s_{11} + 144\,s_{21} - 
  60\,s_0\,s_{001} + 216\,s_{101} + 144\,s_{0001}
\end{eqnarray*}

}




\section{Multisingularities of hypersurfaces}

\subsection{Isolated hypersurface singularities\\ 
and Legendre characteristic classes}

Almost all results concerning the Thom polynomials of singularities and
multisingularities of maps can be translated to the parallel theory of
isolated hypersurface singularities. The only difference is that the
ring $\cC$ is replaced by the ring $\cL$ of universal Legendre
characteristic classes. Here we briefly review the definitions, for the
details see~\cite{K4}.


\ex Definition. The map $M\to N$ is called {\em Legendrian}, if it can
be represented as a composition $M\to PT^*N\to N$ where the first map
is a Legendrian embedding and the second map is the natural
projection.

In particular, $\dim N-\dim M=1$. The Legendre maps form a specific
class of maps those generic singularities are different from those
for the usual maps. For the definition of a Legendre
embedding, see~\cite{AG,AGLV}. For the applications it is sufficient to
consider the following example.

\ex Example. Let $\pi:W\to N$ be a holomorphic locally trivial
fibration, $H\subset W$ be a smooth hypersurface. Denote by $M\subset
H$ the locus of point at which $H$ is tangent to the fibers of $\pi$.
Assume that $M$ is smooth. More precisely, $M$ is given as the zero
set of a section of the bundle $\Hom(V,I)$ on $H$, where $V$ is the
subbundle of $T_HW$ of vectors tangent to the fibers of $\pi$ and $I$
is the normal line bundle of $H$. We assume that this section is
transversal to the zero section, and so by implicit function theorem
$M$ is smooth and $\dim M=\dim N-1$. The Legendre map of this example
is the restriction to $M$ of the projection $\pi$.

The points of the source of a Legendre map are classified according to
the isomorphism classes of isolated hypersurface singularities. In the
example above we associate with a point $x\in M\subset H$ the singularity
of the hypersurface on the fiber $\pi^{-1}\pi(x)$ cut out by the
hypersurface $H$. Respectively, the points of the target manifold $N$
are classified by the isomorphism classes of multisingularities of
hypersurfaces. The beginning of the hierarchy of the isolated
hypersurface singularities is represented on the following diagram
{\renewcommand{\arraystretch}{1.8}
$$\begin{picture}(0,0)
\multiput(136,3)(31,0){4}{\vector(-1,1){15}}
\multiput(198,-24)(31,0){2}{\vector(-1,1){15}}
\multiput(198,-18)(31,0){2}{\vector(-1,2){15}}
\put(229,-45){\vector(-1,1){15}}
\end{picture}
\begin{array}{cc@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c}
\codim&0&&1&&2&&3&&4&&5&&6\\
&A_1&\ot&A_2&\ot&A_3&\ot&A_4&\ot&A_5&\ot&A_6&\ot&A_7\\
&&&&&&&D_4&\ot&D_5&\ot&D_6&\ot&D_7\\
&&&&&&&&&&&E_6&\ot&E_7\\
&&&&&&&&&&&&&P_8
\end{array}$$}

Different multisingularity loci $M(\ua)\subset M$, $N(\ua)\subset N$ for
multisingularity classes $\ua=(\a_1,\dots,\a_r)$ and their Poincar\'e
dual cohomology classes $\om_\ua$, $\on_\ua$ are defined similarly to the
case of usual maps.

In the case of Legendre maps the ring of universal characteristic
classes is defined in a different way comparing with the case of usual
maps.

\ex Definition. {The ring $\cL$ of {\em universal Legendre characteristic
classes\/} is the quotient ring of polynomials in variables
$u,a_1,a_2,\dots$, $\deg u=1$, $\deg a_i=i$, over the ideal generated
by the relations obtained as homogeneous components of the equality:
\begin{equation}
(1+a_1+a_2+\dots)
\(1-\frac{a_1}{1+u}+\frac{a_2}{(1+u)^2}-\dots\)=1.
\label{rel}
\end{equation}
These relations allow to expand all powers of the variables $a_i$. The
additive basis of $\cL$ is formed by the monomials of the form
$u^ka_1^{i_1}a_2^{i_2}\dots$ with $i_j\in\{0,1\}$.

The ring $\cSL$ is defined as the polynomial ring
  $$\cSL=\Sym\cL=1\oplus\cL\oplus\Sym^2\cL\oplus\dots$$
(with no relations).}

With any Legendre map $f:M\to N$ one associates the characteristic
homomorphism
  $$\cL\to H^*(M).$$
In the case of the example above the corresponding characteristic
classes are defined as follows: $u=c_1(I)$ is the restriction to $M$
of the class of the divisor $H$, and the classes $a_i$ are defined as
$a_i=c_i(f^*TN-TM-I)$. The identity~(\ref{rel}) can be reformulated
as $c(U+U^*{\otimes} I)=1$, where $U$ is the formal difference
$U=f^*TN-TM-I$. To verify this identity is a good exercise on
the theory of characteristic classes.

Applying the Gysin homomorphism $f_*:H^*(M)\to H^*(N)$ we can push the
Legendre characteristic classes to the target manifold $N$. Similarly
to the case of usual maps we get the commutative
diagram
$$\begin{array}{ccccccc}
\cL\otimes\cSL&\too[s]&\cSL&\too[1\otimes\Id]&\cL\otimes\cSL\\
\downarrow&&\downarrow&&\downarrow\\
H^*(M)&\too[f_*]&H^*(N)&\too[f^*]&H^*(M)
\end{array}$$

\subsection{Main Theorem for hypersurface multisingularities}

\th Theorem 4. {For any non-ordered collection $\ua=(\a_1,\dots,\a_r)$ 
of isolated hypersurface singularity classes there exists a
universal `residue' Legendre characteristic class $R_\ua\in\cL$ such that
the multisingularity classes $\om_\ua$, $\on_\ua$ are given by the formulas
of Theorem~{\rm1},
\begin{eqnarray*}
m_{\ua}&=&
\!\!\!\sum_{\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l}\!\!\! 
R_{\ua_{J_1}}\;S_{\ua_{J_2}}\dots S_{\ua_{J_l}},\\
n_{\ua}&=&s(m_{\ua})=
\!\!\!\sum_{\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l}\!\!\! 
S_{\ua_{J_1}}\dots S_{\ua_{J_l}},
\end{eqnarray*}
where $S_\ua=s(R_\ua)$ and the summation is taken over all possible 
partitions of the set $\{1,\dots,r\}$ into a disjoint union of
nonempty subsets $\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l$, $l\ge1$.
The subset containing the first element $1\in\{1,\dots,r\}$ is
denoted by $J_1$.}


\ex Example. Similarly to the case of usual maps we obtain, in
particular, the following {\em multiple point formula for Legendre
maps}:
  \begin{equation}
  \om_r=f^*(\on_{r-1})-\sum_{i=1}^{r-1}p_{r-i}\om_r,
  \label{eq8}\end{equation}
where $p_i\in\cL$ are universal Legendre characteristic
classes, defined by
$$\textstyle 1+p_1t+p_2t^2+\dots=\left(1+\frac1{1!}R_{A_1^2}t+
   \frac1{2!}R_{A_1^3}t^2+\dots\right)^{-1},$$
  $$p_1=u + 3\,a_1,\quad p_2=-4\,\left( u\,a_1 + 3\,a_2 \right) ,
\quad p_3=16\,u^2\,a_1 + 59\,u\,a_2 + 45\,a_1\,a_2 + 78\,a_3,\dots.$$
More examples will be given in Section~\ref{enum}.



\subsection{Computations}

The characteristic classes of monosingularities up to codimension $6$
are computed in~\cite{K4}. These computations are fulfilled both
using the classical approach and the method of Rim\'anyi. The Rim\'anyi's
method is less geometric but it requires less computations and can be
easily adapted to the study the multisingularities. In particular, we
have computed all residue polynomials (and thus, the classes of
multisingularity loci) up to codimension $6$, see~\cite{wwwkaz}. Up
to codimension $4$ these classes are represented in the table below.

{\small\subsubsection*
{Residue classes $R_\ua$ of codimension $\le4$ for Legendre
maps}
\begin{eqnarray*}
R_{A_1}&=&1\\
R_{A_2}&=&a_1\\
R_{A_1^2}&=&-u - 3\,a_1\\
R_{A_3}&=&u\,a_1 + 3\,a_2\\
R_{A_1\,A_2}&=&-6\,( u\,a_1 + 2\,a_2 ) \\
R_{A_1^3}&=&2\,( u^2 + 19\,u\,a_1 + 30\,a_2 ) \\
R_{A_4}&=&u^2\,a_1 + 4\,u\,a_2 + 3\,a_1\,a_2 + 6\,a_3\\
R_{D_4}&=&-( u\,a_2 )  + a_1\,a_2 - 2\,a_3\\
R_{A_1\,A_3}&=&-4\,( 
      2\,u^2\,a_1 + 5\,u\,a_2 + 6\,a_1\,a_2 + 3\,a_3 ) \\
R_{A_2^2}&=&-3\,( 
      3\,u^2\,a_1 + 8\,u\,a_2 + 7\,a_1\,a_2 + 6\,a_3 ) \\
R_{A_1^2\,A_2}&=&24\,( 
      3\,u^2\,a_1 + 7\,u\,a_2 + 6\,a_1\,a_2 + 3\,a_3 ) \\
R_{A_1^4}&=&-6\,( 
      u^3 + 111\,u^2\,a_1 + 239\,u\,a_2 + 171\,a_1\,a_2 + 
        78\,a_3 ) \\
R_{A_5}&=&u^3\,a_1 - 4\,u^2\,a_2 + 16\,u\,a_1\,a_2 - 
    12\,u\,a_3 + 27\,a_1\,a_3 + 6\,a_4\\
R_{D_5}&=&-2\,( 
      2\,u^2\,a_2 - 2\,u\,a_1\,a_2 + 7\,u\,a_3 - 3\,a_1\,a_3 + 
        6\,a_4 ) \\
R_{A_1\,A_4}&=&-10\,( 
      u^3\,a_1 - 4\,u^2\,a_2 + 14\,u\,a_1\,a_2 - 16\,u\,a_3 + 
        21\,a_1\,a_3 - 6\,a_4 ) \\
R_{A_1\,D_4}&=&4\,( 
      5\,u^2\,a_2 - 5\,u\,a_1\,a_2 + 16\,u\,a_3 - 6\,a_1\,a_3 + 
        12\,a_4 ) \\
R_{A_2\,A_3}&=&-6\,( 
      2\,u^3\,a_1 - 10\,u^2\,a_2 + 28\,u\,a_1\,a_2 - 39\,u\,a_3 + 
        39\,a_1\,a_3 - 18\,a_4 ) \\
R_{A_1^2\,A_3}&=&2\,( 
      56\,u^3\,a_1 - 220\,u^2\,a_2 + 684\,u\,a_1\,a_2 - 
        951\,u\,a_3 + 891\,a_1\,a_3 - 522\,a_4 ) \\
R_{A_1\,A_2^2}&=&18\,( 
      7\,u^3\,a_1 - 20\,u^2\,a_2 + 74\,u\,a_1\,a_2 - 96\,u\,a_3 + 
        95\,a_1\,a_3 - 50\,a_4 ) \\
R_{A_1^3\,A_2}&=&-48\,( 
      28\,u^3\,a_1 - 55\,u^2\,a_2 + 250\,u\,a_1\,a_2 - 
        318\,u\,a_3 + 300\,a_1\,a_3 - 180\,a_4 ) \\
R_{A_1^5}&=&24\,( 
      u^4 + 671\,u^3\,a_1 - 701\,u^2\,a_2 + 4863\,u\,a_1\,a_2 - 
        5844\,u\,a_3 + 5490\,a_1\,a_3 - 3420\,a_4 ) 
\end{eqnarray*}}





\subsection{Application to enumerative geometry}\label{enum}



The enormous number (several hundred) of concrete enumerative results
following from the formulas of this paper is impossible to present
here because of the huge volume of the output (see~\cite{wwwkaz}). 
We shall list only few examples.


\subsubsection{Enumerating singular curves on surfaces}

The simplest application of the theory described above is
the enumeration of singular curves on a surface. Consider
a linear system $L$ of dimension $n$ on a smooth complex complete surface
$S$. Then our main formula predicts the number of singular curves of
this system having the prescribed collection $\ua$ of singularities
and passing through $n-\codim\ua$ general points. The parameter space
$N=\C P^n$ of this problem is the parameter space of the system. The
bundle $W\to N$ is the trivial bundle $S\times N\to N$, $H\subset
S\times N$ is the incidence hypersurface, the manifold of all
points of all curves of the system, $M\subset H$ is the locus of
singular points of the curves. The Legendre characteristic classes
related to this problem are the following. Let $h\in H^2(S)$ be the first
Chern class of the system, $1+c_1+c_2=c(T^*S)$, and $t$ be the class of
a hyperplane on $N=\C P^n$. Then 
$$u=c_1(I)=[H]=t+h,\quad a=c(T^*S\otimes I-TS)=
\frac{(1+t+h)^2+(1+t+h)c_1+c_2}{1-c_1+c_2},$$ that is
$$a_1= 2\,( h + t + c_1 ) ,\quad
a_2=( h + t + c_1 ) ( h + t + 2\,c_1 ) ,\quad
a_3=    t\,( 2\,h\,c_1 + t\,c_1 + 3\,c_1^2 - 2\,c_2 ) ,$$
$$a_4=  t^2\,( c_1^2 - c_2 ),\quad a_k=0\hbox{ for
}k\ge5.$$ 
The homomorphism $f_*:H^*(M)\to H^*(N)$ can be decomposed
as $f_*=p_*i_*$ with $i:M\to S\times N$ the embedding and
$p:S\times N\to N$ the projection to the second factor. Then $i_*$
is given by the multiplication by the class of $M$ in $S\times N$,
 \begin{eqnarray*}
[M]&=&i_*(1)=[H]\cdot c_2(T^*S\otimes I)=(t+h)((t+h)^2+c_1(t+h)+c_2))\\
&=&t^3 + t^2\,( 3\,h + c_1 ) + t\,( 3\,h^2 + 2\,h\,c_1 + c_2 )
\end{eqnarray*} 
and $p_*$ is defined as follows. It commutes with $t$; on the
monomials of degree $2$ in $h,c_1,c_2$ it is given by the
corresponding characteristic numbers,
$$\begin{array}{l@{\qquad}l}
d:=p_*(h^2)=L\cdot L,&
k:=p_*(hc_1)=L\cdot K_S,\\
s:=p_*(c_1^2)=K_S\cdot K_S,&
x:=p_*(c_2)=c_2\smallfrown[S];
\end{array}$$
and $p_*$ vanishes on other monomials in $h,c_1,c_2$. As a consequence
we get that for any multisingularity $\ua$ the class $S_\ua=f_*R_\ua$
is a universal linear combination of the numbers $d,k,s,x$ multiplied
by $t^{\codim\ua}$ and the formula~(\ref{eq2}) of Theorem~1 reduces to
the following result refining the conjecture of G\"ottsche~\cite{G} and the
known results due to Kleiman and Piene~\cite{KP1,KP2}.

\th Theorem. For any plane curve multisingularity type
$\ua=(\a_1,\dots,\a_r)$ there exists a universal linear combination
$S_\ua$ of the numbers $d,k,s,x$ such that the number $N_\ua$ of
singular curves of a `sufficiently generic' system $L$ with the given
collection $\ua$ of singularities and passing through a given
$n-\codim\ua$ points is given by the formula~{\rm(\ref{eq2})} of
Theorem~{\rm1:}
\begin{eqnarray*}
N_{\ua}&=&\frac1{\#\Aut(\ua)}
\!\!\!\sum_{\{1,\dots,r\}=J_1\sqcup\dots\sqcup J_l}\!\!\! 
S_{\ua_{J_1}}\dots S_{\ua_{J_l}}.
\end{eqnarray*}

The genericity condition on the system is formulated as follows.
Denote by $N_{\rm reg}\subset N$ the subset of curves $C$ in the system that have
finite number of singular points and satisfying the following
transversality condition: the deformation of $C$ induced by the system
induces a versal deformation of the multigerm of singularities of $C$.
Denote by $\S=N\setminus N_{\rm reg}$ its complement.
Then the formula of the theorem holds for all classes $\ua$ of
codimension 
$$\codim\ua<\codim\S.$$

For multisingularities of small codimension the numbers $S_\ua$ are
presented below. These expressions agree with those obtained by
Kleiman and Piene~\cite{KP1,KP2} for the nodal multisingularities
$A_1^r$. (the complete list of these numbers for all
multisingularities up to codimension $7$ can be found
in~\cite{wwwkaz}).
{\small
\begin{eqnarray*}
    &&\quad\codim=1 :\\
S_{A_1} &=&3\,d + 2\,k + x\\
    &&\quad\codim=2 :\\
S_{A_2} &=&12\,d + 12\,k + 2\,s + 2\,x\\
S_{A_1^2} &=&-42\,d - 39\,k - 6\,s - 7\,x\\
    &&\quad\codim=3 :\\
S_{A_3} &=&50\,d + 64\,k + 17\,s + 5\,x\\
S_{A_1\,A_2} &=&-240\,d - 288\,k - 72\,s - 24\,x\\
S_{A_1^3} &=&1380\,d + 1576\,k + 376\,s + 138\,x\\
    &&\quad\codim=4 :\\
S_{A_4} &=&180\,d + 280\,k + 100\,s \\
S_{D_4} &=&15\,d + 20\,k + 5\,s + 5\,x\\
S_{A_1\,A_3} &=&-1260\,d - 1820\,k - 596\,s - 60\,x\\
S_{A_2^2} &=&-1260\,d - 1800\,k - 588\,s - 48\,x\\
S_{A_1^2\,A_2} &=&9000\,d + 12360\,k + 3864\,s + 456\,x\\
S_{A_1^4} &=&-72360\,d - 95670\,k - 28842\,s - 3888\,x\\
    &&\quad\codim=5 :\\
S_{A_5} &=&630\,d + 1140\,k + 498\,s - 60\,x\\
S_{D_5} &=&84\,d + 132\,k + 44\,s + 20\,x\\
S_{A_1\,A_4} &=&-5460\,d - 9240\,k - 3740\,s + 200\,x\\
S_{A_1\,D_4} &=&-420\,d - 624\,k - 196\,s - 100\,x\\
S_{A_2\,A_3} &=&-6300\,d - 10332\,k - 4044\,s + 60\,x\\
S_{A_1^2\,A_3} &=&52920\,d + 84180\,k + 31816\,s + 240\,x\\
S_{A_1\,A_2^2} &=&53676\,d + 84456\,k + 31716\,s + 72\,x\\
S_{A_1^3\,A_2} &=&-505008\,d - 770112\,k - 279792\,s - 5616\,x\\
S_{A_1^5} &=&5225472\,d + 7725168\,k + 2723400\,s + 84384\,x\\
    &&\quad\dots\\
S_{A_1^6} &=&-481239360\,d - 778065120\,k - 308078520\,s + 7918560\,x\\
S_{A_1^7} &=&53917151040\,d + 93895251840\,k + 40747613760\,s - 2465471520\,x
\end{eqnarray*}
}

For example, in the particular case of the  complete system of degree
$m$ plane curves we have $S=\C P^2$, $n=\frac{(m+1)(m+2)}2-1$,
$d=m^2$, $k=3m$, $s=9$, $x=3$. The formula of the theorem holds in this case
presumably for
$$\codim\ua\le2\,(m-1)$$ 
(similar bound was conjectured in~\cite{G}.)
The restriction $\codim\S\le2m-1$ follows from the
fact that $\S$ contains the codimension $2m-1$ subvariety of curves
that split into a curve of degree $m-2$ and a double line. Such curves
have non-isolated singularities that is singularities which are
infinitely degenerate from the point of view of singularity theory.

\ex Example. For degree $m=3$ plane curves the formulas above are
valid for $\codim\ua\le4$ and we get:
$$\begin{array}{c|cccccccccccccccccccc}
\ua&A_1&A_2&A_1^2&A_3&A_1A_2&A_1^3&
  A_4&D_4&A_1A_3&A_2^2&A_1^2A_2&A_1^4\\\hline
N_\ua&12&24&21&42&0&15&0&15&0&0&0&0
\end{array}$$
All these numbers can easily be verified by elementary computations.
For example, $N_{A_1}=12$ is the number of rational curves of degree
$3$ passing through generic $8$ points, $N_{D_4}=15$ is the number
of configurations of three lines meeting at one point and passing
through $5$ fixed points of the plane etc.

\ex Example. The number of degree $4$ plane curves with $3$ nodes passing
through $11$ generic points is $N_{A_1^3}|_{m=4}=675$. This number
includes ${11\choose2}=55$ curves that split into a line and a
cubic and the remaining number $620$ of rational curves.

\ex Example. Substituting $m=5$ to the formula for $N_{A_1^6}$ we get
the number $109781$ of curves of degree $5$ with $6$ nodes through
generic $14$ points. For such a curve $C$ we have one of the following
three possibilities: a) $C$ splits into a cubic
and a quadric; b) $C$ splits into a line and a degree $4$ curve with
$2$ nodes; c) $C$ is an irreducible rational curve. The number of
curves of type a) is ${14\choose5}=2002$. The number of curves of type b)
is $225{14\choose2}=20475$ (the number $225=\deg A_1^2|_{m=4}$ of
degree $4$ curves with $2$ nodes through $12$ points is obtained from the
same table). Therefore, the number of rational curves of type c) is
$109781-2002-20475=87304$ that agrees with the well known computations
of~\cite{KM}.

More general, consider the numbers $N_{m,g}$ of irreducible plane
curves of degree $m$ and genus $g$ passing through general $3m-1-g$
points. These numbers are studied in Gromov-Witten theory and related
to the numbers $M_{m,k}$ of all degree $m$ plane curves with
$k$ nodes passing through general $\frac{m(m+3)}2-k$ points by
the following relation on their generating functions that generalizes
the computations above:
$$1+\sum_{m,k}M_{m,k}
\frac{x^my^{\frac{m(m+3)}2-k}}{(\frac{m(m+3)}2-k)!}=
\exp\(\sum_{m,g}N_{m,g}\frac{x^my^{3m+g-1}}{(3m+g-1)!}\).$$

Unfortunately, our method for computing the numbers $M_{m,k}$ and
thus the numbers $N_{m,g}$ provides the correct answer only for
$k\le2(m-1)$ and for greater $k$ one should take into account the
contribution of non-isolated singularities.


\bigskip The problem considered above is a particular case of the
following more general one: given an $n$-dimensional linear system on
a smooth projectile manifold  $V$, what is the degree of the
collection of divisors of this system having isolated singular points
of the given multisingularity type?  In other words,  what is the
degree of the variety of hyperplanes in $\C P^{n}$ having prescribed
multisingularity type of the intersection with a given subvariety
$V\subset \C P^n$? (In the problem of the enumeration of degree $m$
plane curves the question is about the Veronese embedding $\C P^2\to\C
P^{\frac{(d+1)(d+2)}2-1}$.)  The source manifold $M$ of the
underlying Legendre map is the projectivized conormal bundle of $V$
and the target manifold is the projective space $\C P^{n*}$ of all
projective hyperplanes in $\C P^n$. The Legendre characteristic
classes in this situation is easy to compute. For example, if $V$ is
a hypersurface of degree $d$ then $M=V$, $u=[V]=d t$,
$a=(\frac{1+(d-1)t}{1+t})^{n+1}$,  where $t=c_1(\cO(1))$ is the class
of the hyperplane section. The numbers of various singular hyperplane
sections of a generic degree $d$ hypersurface in $\C P^n$ for small
$n$ is presented in the lists below\footnote{The dots denote some
missing terms. They are available and commented out in the source
\LaTeX\/ file, see also~\cite{wwwkaz} where one can find expressions
for the degrees of the corresponding singularity loci in case if
their dimension is positive.\label{ftnt}}.


\subsubsection{Multitangent lines to a curve in $\C P^2$}
{\small\begin{eqnarray*}
\#A_2&=&3\,d\,( d - 2 )\\ 
\#A_1^2&=&\frac{1}{2}\,d\,( d - 3 ) ( d - 2 ) ( d + 3 ) 
\end{eqnarray*}}

\subsubsection{Multitangent planes to a surface in $\C P^3$}
{\small\begin{eqnarray*}
\#A_3&=&2\,d\,( 11\,d - 24 ) ( d - 2 )\\ 
\#A_1A_2&=&4\,d\,( d - 3 ) ( d - 2 ) ( d^3 + 3\,d - 16 )\\ 
\#A_1^3&=&\frac{1}{6}\,d\,( d - 2 ) ( d^7 - 4\,d^6 + 7\,d^5 - 
  45\,d^4 + 114\,d^3 - 111\,d^2 + 548\,d - 960 )
\end{eqnarray*}}

\subsubsection{Multitangent hyperplanes to a hypersurface in $\C
P^4$}
{\small\begin{eqnarray*}
\#A_4&=&5\,d\,( 17\,d - 36 ) ( 3\,d - 7 ) ( d - 2 )\\ 
\#D_4&=&20\,d\,{(d - 2 ) }^3\\ 
\#A_1A_3&=&5\,d\,( d - 2 ) ( 7\,d^6 - 43\,d^5 + 102\,d^4 - 
  118\,d^3 - 245\,d^2 + 1309\,d - 1404 )\\ 
\#A_2^2&=&\frac{5}{2}\,d\,( d - 3 ) ( d - 2 ) ( 
  5\,d^5 - 15\,d^4 + 25\,d^3 - 5\,d^2 - 273\,d + 452 ) \\ 
\#A_1^2A_2&=&\frac{5}{2}\,d\,( d - 2 ) ( 
  d^{10} - 8\,d^9 + 28\,d^8 - 
  56\,d^7 - 6\,d^6 + 398\,d^5 - 1028\,d^4 \\&&\qquad{}+ 
  1196\,d^3 + 1317\,d^2 - 8034\,d + 8424 ) \\ 
\#A_1^4&=&\frac{1}{24}\,d\,( d - 2 ) ( 
  d^{14} - 10\,d^{13} + 46\,d^{12} - 
  128\,d^{11} + 143\,d^{10} + 442\,d^9 
% - 2320\,d^8 + 4984\,d^7 - 2401\,d^6 - 17990\,d^5 
+\dots\\&&\qquad{}
+ 50434\,d^4 - 59080\,d^3 - 40415\,d^2 + 287430\,d - 296280 )
\end{eqnarray*}}

\subsubsection{Multitangent hyperplanes to a hypersurface in $\C
P^5$}
{\small\begin{eqnarray*}
\#A_5&=&6\,d\,( d - 2 ) ( 746\,d^3 - 5012\,d^2 + 11207\,d -  8340 )\\ 
\#D_5&=&140\,d\,( 4\,d - 9 ) {( d - 2 ) }^3\\ 
\#A_1A_4&=&4\,d\,( d - 2 ) ( 114\,d^8 - 1068\,d^7 + 4173\,d^6 - 
  8835\,d^5 + 10980\,d^4 
\\&&\qquad{}
  - 16324\,d^3 + 57313\,d^2 - 118138\,d + 85140 )\\ 
\#A_1D_4&=&35\,d\,{( d - 2 ) }^3( d^6 - 5\,d^5 + 10\,d^4 - 
  10\,d^3 + 5\,d^2 -  69\,d + 144 )\\ 
\#A_2A_3&=&18\,d\,( d - 2 ) (17\,d^8 - 155\,d^7 + 592\,d^6 - 
  1230\,d^5 + 1505\,d^4 - 3091\,d^3 
\\&&\qquad{}
  + 13376\,d^2 -  27964\,d + 20040 )\\ 
\#A_1^2A_3&=&\frac{3}{2}\,d\,( d - 2 ) ( 
  17\,d^{13} - 206\,d^{12} + 1125\,d^{11} - 
  3660\,d^{10} + 7890\,d^9 
+\dots\\&&\qquad{}
%- 13999\,d^8 + 32409\,d^7 -  85486\,d^6 + 163435\,d^5 
  - 196245\,d^4 + 233696\,d^3 - 643864\,d^2 + 1256408\,d - 882000 ) \\ 
\#A_1A_2^2&=&18\,d\,( d - 2 ) ( d^{13} - 12\,d^{12} + 65\,d^{11} - 
  210\,d^{10} + 450\,d^9 - 818\,d^8 
+\dots\\&&\qquad{}
%+ 2042\,d^7 -  5603\,d^6 + 10790\,d^5 
  - 12940\,d^4 + 16853\,d^3 - 51677\,d^2 + 102274\,d - 71820 )\\ 
\#A_1^3A_2&=&d\,( d - 2 ) ( d^{18} - 15\,d^{17} + 105\,d^{16} - 
  455\,d^{15} + 1365\,d^{14} - 3168\,d^{13} 
+\dots\\&&\qquad{}
%+ 6979\,d^{12} - 17100\,d^{11} + 40815\,d^{10} - 
% 78535\,d^9 + 124481\,d^8 - 224857\,d^7 + 494851\,d^6 - 
%  886310\,d^5 + 1039135\,d^4 
  - 1129608\,d^3 + 2730828\,d^2 - 5141832\,d + 3548160 )\\ 
\#A_1^5&=&\frac{1}{120}\,d\,( d - 2 ) ( 
  d^{23} - 18\,d^{22} + 154\,d^{21} - 
  832\,d^{20} + 3181\,d^{19} - 9332\,d^{18} 
+\dots\\&&\qquad{}
%+ 23306\,d^{17} -  56258\,d^{16} + 137704\,d^{15} - 
% 315702\,d^{14} +  632037\,d^{13} -  1167746\,d^{12} + 
% 2276543\,d^{11} - 4606484\,d^{10} + 8183892\,d^9 - 12182630\,d^8 + 
% 19262625\,d^7 - 37322080\,d^6 + 63347155\,d^5 - 72821310\,d^4 
+ 73475394\,d^3 -  156527928\,d^2 + 284455368\,d -  193415040 )
\end{eqnarray*}}

\subsubsection{Multitangent hyperplanes to a hypersurface in $\C
P^6$}
{\small\begin{eqnarray*}
\#A_6&=&7\,d\,( d - 2 ) ( 15926\,d^4 - 142734\,d^3 + 479213\,d^2 - 
  714314\,d + 398856 )\\ 
\#D_6&=&224\,d\,{( d - 2 ) }^3( 46\,d^2 - 209\,d + 237 )\\ 
\#E_6&=&42\,d\,{( d - 2 ) }^3( 97\,d^2 - 426\,d + 468 )\\ 
\#A_1A_5&=&7\,d\,( d - 2 ) ( 1240\,d^{10} - 15630\,d^9 + 85751\,d^8 - 
  268904\,d^7 + 531693\,d^6
+\dots\\&&\qquad{}
% - 688330\,d^5 + 448411\,d^4 
+ 881076\,d^3 - 3841807\,d^2 + 5752566\,d - 3162744 )\\ 
\#A_1D_5&=&56\,d\,{
( d - 2 ) }^3( 19\,d^8 - 156\,d^7 + 537\,d^6 - 
  1010\,d^5 + 1125\,d^4 - 744\,d^3 
\\&&\qquad{}
- 1617\,d^2 + 8182\,d - 8952 )\\ 
\#A_2A_4&=&7\,d\,( d - 2 ) ( 742\,d^{10} - 9135\,d^9 + 49070\,d^8 - 
  151025\,d^7 + 293748\,d^6 
+\dots\\&&\qquad{}
%-  374857\,d^5 + 165220\,d^4 
+ 1132215\,d^3 - 4197325\,d^2 + 6167502\,d - 3364920 )\\ 
\#A_2D_4&=&56\,d\,{( d - 2 ) }^3( 7\,d^8 - 56\,d^7 + 189\,d^6 - 
  350\,d^5 + 385\,d^4 - 252\,d^3 
+\dots\\&&\qquad{}
- 1205\,d^2 +  5626\,d - 6144 )\\ 
\#A_3^2&=&\frac{7}{2}\,d\,( d - 2 ) ( 700\,d^{10} - 8540\,d^9 + 45507\,d^8 - 
  139076\,d^7 + 268849\,d^6 
+\dots\\&&\qquad{}
% -  341250\,d^5 + 117469\,d^4 
+ 1299760\,d^3 - 4672757\,d^2 + 6818290\,d - 3701976 ) \\ 
\#A_1^2A_4&=&\frac{7}{2}\,d\,( d - 2 ) ( 106\,d^{16} - 1729\,d^{15} + 
  12972\,d^{14} - 59386\,d^{13} + 185482\,d^{12} 
+\dots\\&&\qquad{}
%- 418407\,d^{11} +  682936\,d^{10} - 641386\,d^9 - 508952\,d^8 + 
% 3632163\,d^7 - 8009538\,d^6 + 10589188\,d^5 -  7689796\,d^4 
- 6493787\,d^3 + 35146830\,d^2 - 52506056\,d + 28436160 ) \\ 
\#A_1^2D_4&=&28\,d\,{( d - 2 ) }^3( d^{14} - 12\,d^{13} + 66\,d^{12} - 
  220\,d^{11} + 495\,d^{10} - 792\,d^9 
+\dots\\&&\qquad{}
% + 742\,d^8 + 678\,d^7 - 4503\,d^6  
+ 9090\,d^5 - 10224\,d^4 + 6750\,d^3 + 8471\,d^2 -  46098\,d + 49500 )\\ 
\#A_1A_2A_3&=&7\,d\,( d - 2 ) ( 70\,d^{16} - 1127\,d^{15} + 
  8358\,d^{14} - 37870\,d^{13} + 117194\,d^{12} 
+\dots\\&&\qquad{}
% - 262185\,d^{11} + 420310\,d^{10} - 342126\,d^9 - 599842\,d^8 + 
% 3093897\,d^7 - 6557374\,d^6 +  8549732\,d^5 - 5773318\,d^4 
- 8489681\,d^3 + 38495186\,d^2 - 56595816\,d + 30458304 )\\ 
\#A_2^3&=&\frac{7}{6}\,d\,( d - 2 ) ( 
 49\,d^{16} - 784\,d^{15} + 5782\,d^{14} - 
  26068\,d^{13} + 80311\,d^{12} 
+\dots\\&&\qquad{}
% - 178948\,d^{11} + 284676\,d^{10} - 218841\,d^9 - 470001\,d^8 + 
%  2269253\,d^7 - 4741520\,d^6 + 6132413\,d^5 - 3735995\,d^4 
- 9304861\,d^3 + 37829206\,d^2 - 55148904\,d + 29638080 ) \\ 
\#A_1^3A_3&=&\frac{7}{6}\,d\,( d - 2 ) ( 10\,d^{22} - 201\,d^{21} + 
  1908\,d^{20} - 11373\,d^{19} + 47736\,d^{18} 
+\dots\\&&\qquad{}
% - 149940\,d^{17} + 363012\,d^{16} -  666858\,d^{15} + 799650\,d^{14} - 
%  15554\,d^{13} - 2847240\,d^{12} + 8407224\,d^{11} - 14974646\,d^{10} + 
%  16463036\,d^9 - 943452\,d^8 - 42526574\,d^7 + 102463652\,d^6 - 
%  137371749\,d^5 + 102778412\,d^4 
+ 56118029\,d^3 -  358281042\,d^2 + 532314216\,d -  284437440 ) \\ 
\#A_1^2A_2^2&=&\frac{7}{4}\,d\,( d - 2 ) ( 7\,d^{22} - 140\,d^{21} + 
  1323\,d^{20} - 7854\,d^{19} + 32844\,d^{18} 
+\dots\\&&\qquad{}
%- 102816\,d^{17} + 247958\,d^{16} - 451507\,d^{15} + 521490\,d^{14} + 
%   87914\,d^{13} - 2226268\,d^{12} + 6337599\,d^{11} - 11129724\,d^{10} + 
%   11824386\,d^9 + 1302312\,d^8 - 37396247\,d^7 + 86981989\,d^6 - 
%   115525152\,d^5 + 83953803\,d^4 
 + 64936185\,d^3 - 356391246\,d^2 + 525011976\,d - 280089792 ) \\ 
\#A_1^4A_2&=&\frac{7}{24}\,d\,( d - 2 ) ( d^{28} - 24\,d^{27} + 
  276\,d^{26} - 2024\,d^{25} + 10626\,d^{24}
+\dots\\&&\qquad{}
%  - 42504\,d^{23} + 134296\,d^{22} -  340116\,d^{21} + 678987\,d^{20} - 
%   972740\,d^{19} + 563448\,d^{18} + 1873536\,d^{17} - 7860616\,d^{16} + 
%   17287976\,d^{15} - 24872172\,d^{14} + 16431960\,d^{13} + 28587427\,d^{12} - 
%   120111384\,d^{11} + 230184424\,d^{10} - 269176824\,d^9 + 80781174\,d^8 + 
%   462278544\,d^7 - 1203527324\,d^6 + 1632326892\,d^5 - 1240565023\,d^4 - 
  494311612\,d^3 + 3664055868\,d^2 - 5430061008\,d + 2876929920 ) \\ 
\#A_1^6&=&\frac{1}{720}\,d\,( d - 2 ) (  d^{34} - 28\,d^{33} + 
  379\,d^{32} - 3302\,d^{31} + 20801\,d^{30}  
+\dots\\&&\qquad{}
%- 100904\,d^{29} + 391637\,d^{28} - 1243976\,d^{27} + 3258773\,d^{26} - 
%   6947804\,d^{25} + 11367707\,d^{24} - 11171186\,d^{23} - 
%   6991007\,d^{22} + 64122816\,d^{21} - 175223643\,d^{20} + 
%   311944164\,d^{19} - 345684657\,d^{18} + 17824836\,d^{17} + 
%   965696817\,d^{16} - 2624426226\,d^{15} + 4193597763\,d^{14} - 
%   3700626784\,d^{13} - 1563846193\,d^{12} + 12817373224\,d^{11} - 
%   26582836185\,d^{10} + 32627381860\,d^9 - 14788060935\,d^8 - 
%   38654085750\,d^7 + 111158033947\,d^6 - 153085008888\,d^5 + 
%   118578938359\,d^4 + 29125471228\,d^3 
- 287028053004\,d^2 + 426393893520\,d - 224430998400 ) 
\end{eqnarray*}}

The formulas for lines on the plane and for planes in the $3$-space
are classical results of Pl\"ucker and Salmon. It seems that similar
formulas for the case of hyperplanes in $\C P^n$, $n\ge 4$, have not been
known before. The simplest case is {\bf 495} hyperplanes in $\C P^4$
tangent to a degree $3$ hypersurface at $4$ points.

\bigskip
A very close problem is the following.
Consider a smooth projective hypersurface $V\subset\C P^n$ of degree
$d$. The theory developed above allows to compute the number of
$k$-dimensional projective subspaces $K$ with prescribed collection of
singularities of the intersection $K\cap V$. The source manifold of
the underlying Legendre map is $M=G_k(TV)$, the Grassmannian bundle
over $V$ of $k$-dimensional tangent subspaces and the target manifold
is the Grassmannian $N=G_{k+1,n+1}$ of all $k$-dimensional projective
subspaces in $\C P^n$. The associated Legendre characteristic classes
on $M$ and the Gysin homomorphism $f_*:H^*(M)\to H^*(N)$ can be easily
computed using various adjunction formulas. Some of the results of these
computations are presented below\footnote{See the footnote on
page~\pageref{ftnt}.}. 


\subsubsection{$k=1$, $n=3$ (lines in the space)}
{\small\begin{eqnarray*}
\#A_4&=&5\,d\,( 7\,d - 12 ) ( d - 4 )\\ 
\#A_1A_3&=&2\,d\,( d - 5 ) ( 3\,d - 5 ) ( d - 4 ) ( d + 6 )\\ 
\#A_2^2&=&\frac{1}{2}\,d\,( d - 5 ) ( d - 4 ) 
  ( d^3 + 3\,d^2 + 29\,d -  60 ) \\ 
\#A_1^2A_2&=&\frac{1}{2}\,d\,( d - 6 ) ( d - 5 ) ( d - 4 ) 
  ( d^3 + 9\,d^2 + 20\,d -  60 ) \\ 
\#A_1^4&=&\frac{1}{12}\,d\,( d - 7 ) ( d - 6 ) 
  ( d - 5 ) ( d - 4 ) ( d^3 + 6\,d^2 + 7\,d - 30 )
\end{eqnarray*}}

\subsubsection{$k=1$, $n=4$ (lines in $\C P^4$)}
{\small\begin{eqnarray*}
\#A_6&=&35\,d\,( 7\,d - 12 ) ( 3\,d - 10 ) ( d - 6 )\\ 
\#A_1A_5&=&5\,d\,( d - 7 ) ( d - 6 ) ( 17\,d^3 + 69\,d^2 - 662\,d + 840 )\\ 
\#A_2A_4&=&5\,d\,( d - 7 ) ( d - 6 ) 
  ( 2\,d^4 + 6\,d^3 + 91\,d^2 -  678\,d + 840 )\\ 
\#A_3^2&=&\frac{1}{2}\,d\,( d - 7 ) ( d - 6 ) 
  ( d^5 + d^4 + 65\,d^3 + 395\,d^2 - 3350\,d + 4200 ) \\ 
\#A_1^2A_4&=&\frac{5}{2}\,d\,( d - 8 ) ( d - 7 ) ( d - 6 ) 
  ( 2\,d^4 + 23\,d^3 + 13\,d^2 -  598\,d + 840 ) \\ 
\#A_1A_2A_3&=&d\,( d - 8 ) ( d - 7 ) ( d - 6 )
  ( d^5 + 11\,d^4 + 95\,d^3 + 115\,d^2 - 3030\,d + 4200 )\\ 
\#A_2^3&=&\frac{1}{6}\,d\,( d - 8 ) ( d - 7 ) ( d - 6 ) 
  ( d^5 + 21\,d^4 + 40\,d^3 +  225\,d^2 - 3110\,d +  4200 ) \\ 
\#A_1^3A_3&=&\frac{1}{6}\,d\,( d - 9 ) ( d - 8 ) ( d - 7 ) 
  ( d - 6 ) ( d^5 + 21\,d^4 + 125\,d^3 - 165\,d^2
\\&&\qquad{}
 - 2710\,d + 4200 ) \\ 
\#A_1^2A_2^2&=&\frac
{1}{4}\,d\,( d - 9 ) ( d - 8 ) ( d - 7 ) ( d - 6 ) 
  ( 2\,d^5 + 22\,d^4 + 105\,d^3 
\\&&\qquad{}
- 115\,d^2 - 2750\,d + 4200 ) \\ 
\#A_1^4A_2&=&\frac{1}{24}\,d\,( d - 10 ) ( d - 9 ) ( d - 8 ) 
  ( d - 7 ) ( d - 6 ) ( 3\,d^5 + 33\,d^4 + 115\,d^3 
\\&&\qquad{}
- 345\,d^2 - 2470\,d + 4200 ) \\ 
\#A_1^6&=&\frac{1}{144}\,d\,( d - 11 ) ( d - 10 ) ( d - 9 ) 
  ( d - 8 ) ( d - 7 ) ( d - 6 ) ( d^5 + 9\,d^4 + 21\,d^3 
\\&&\qquad{}- 105\,d^2 - 446\,d + 840 )
\end{eqnarray*}}

\subsubsection{$k=2$, $n=4$ (planes in $\C P^4$)}
{\small\begin{eqnarray*}
\#A_6&=&35\,d\,( 187\,d^3 - 1495\,d^2 + 3794\,d - 3072 )\\ 
\#D_6&=&10\,d\,( d - 2 ) ( 71\,d^2 - 376\,d + 456 )\\ 
\#E_6&=&15\,d\,( 19\,d - 36 ) ( d - 3 ) ( d - 2 )\\ 
\#A_1A_5&=&5\,d\,( 236\,d^6 - 2116\,d^5 + 7092\,d^4 - 
  22919\,d^3 + 98239\,d^2 - 222282\,d + 172656 )\\ 
\#A_1D_5&=&10\,d\,( d - 2 ) ( 15\,d^5 - 89\,d^4 + 181\,d^3 - 
  917\,d^2 + 3752\,d - 4224 )\\ 
\#A_2A_4&=&5\,d\,( 40\,d^7 - 312\,d^6 + 864\,d^5 - 
  976\,d^4 - 12859\,d^3 + \dots
%+ 98791\,d^2 
- 237618\,d + 184320 )\\ 
\#A_2D_4&=&10\,d\,( d - 4 ) ( d - 2 ) 
  ( d^5 + d^4 + d^3 + 21\,d^2 - 460\,d + 732 )\\ 
\#A_3^2&=&\frac{1}{2}\,d\,( 256\,d^7 - 2336\,d^6 + 8981\,d^5 - 
  19846\,d^4 +\dots 
%- 46625\,d^3 + 531270\,d^2 
- 1317380\,d + 1022520 ) \\ 
\#A_1^2A_4&=&10\,d\,( d - 4 ) ( 10\,d^8 - 56\,d^7 + 117\,d^6 - 
  950\,d^5 +\dots
% + 4164\,d^4 - 8298\,d^3 + 34967\,d^2 
 - 105884\,d + 97860 )\\ 
\#A_1^2D_4&=&\frac{5}{2}\,d\,( d - 2 ) ( 3\,d^8 - 20\,d^7 + 47\,d^6 - 
  319\,d^5 +\dots
% + 1579\,d^4 -  3110\,d^3 + 11726\,d^2 
 - 43272\,d + 46800 ) \\ 
\#A_1A_2A_3&=&2\,d\,( 16\,d^{10} - 137\,d^9 + 499\,d^8 - 
  2274\,d^7 +\dots
% + 6385\,d^6 +  8948\,d^5 - 79258\,d^4 + 474345\,d^3 - 2491730\,d^2 
 +  5600880\,d - 4213440 )\\ 
\#A_2^3&=&\frac{4}{3}\,d\,( d^{11} - 12\,d^{10} + 75\,d^9 - 
  302\,d^8 + 354\,d^7 + \dots
%+  1887\,d^6 - 6895\,d^5 +  7332\,d^4 + 80235\,d^3 - 585025\,d^2 
+ 1356750\,d - 1022400 ) \\ 
\#A_1^3A_3&=&\frac{1}{6}\,d\,( 32\,d^{12} - 316\,d^{11} + 
  994\,d^{10} - 4051\,d^9 +\dots
% + 29144\,d^8 - 95884\,d^7 + 324816\,d^6 - 1771439\,d^5 + 
% 5592454\,d^4 - 15093010\,d^3 + 52462740\,d^2 
 - 107808480\,d + 78883200 ) \\ 
\#A_1^2A_2^2&=&d\,( d - 4 ) ( d^{12} - 4\,d^{11} + 
  5\,d^{10} - 110\,d^9 +\dots
% + 115\,d^8 + 640\,d^7 + 3984\,d^6 + 11456\,d^5 - 
%   114922\,d^4 + 281610\,d^3 - 1738365\,d^2 
 + 5382180\,d - 4849740 )\\ 
\#A_1^4A_2&=&\frac{1}{12}\,d\,( d - 4 ) ( 2\,d^{14} - 10\,d^{13} 
- 18\,d^{12} +\dots 
% - 126\,d^{11} + 1257\,d^{10} - 470\,d^9 + 14726\,d^8 - 94548\,d^7 + 
% 120415\,d^6 - 995344\,d^5 + 4798486\,d^4 - 9948710\,d^3 + 40327980\,d^2 
- 113728680\,d + 99802800 ) \\ 
\#A_1^6&=&\frac{1}{144}\,d\,( d^{17} - 12\,d^{16} + 24\,d^{15} + 
  155\,d^{14} - 405\,d^{13} + 1082\,d^{12} 
+\dots\\&&\qquad{}
%- 18469\,d^{11} + 66446\,d^{10} - 192307\,d^9 + 1242535\,d^8 - 
% 4049006\,d^7 + 11129818\,d^6 - 53664614\,d^5 + 166756120\,d^4 
- 415820104\,d^3 + 1293514896\,d^2 - 2517392160\,d + 1781049600 ) 
\end{eqnarray*}}

These formulas agree with those obtained by Le Barz~\cite{LB},
Colley~\cite{Co1,Co2} on the enumeration of multitangent lines to a
projective subvariety and by Vainsencher~\cite{Vains}, Kleiman and
Piene~\cite{KP1,KP2} on the enumeration of singular curves on a
surface.


\subsection{Concluding remarks}

1) The advantage of the singularity theory approach to the problems
of enumerative geometry is that we can treat various problems in a
unique way. Indeed, enumerating singular curves on a surface or
enumerating multitangent lines to a hypersurface or enumerating
singular hyperplane sections of a given variety are considered here
as {\em different reformulations of the same problem\/} of
enumerating multisingularities of a Legendre map (but, for example,
the singularity theory problem responsible for the enumeration of
multitangent lines to a subvariety of codimension greater than $1$ is
a different one). 

2) The manifolds, bundles etc., appearing in particular geometrical
situations have many characteristic classes. The a priori formulas
similar to those of this paper show that the final result is
formulated in terms of certain combinations of these characteristic
classes. This fact is not always seen in geometrical problems and may
simplify considerably finding the desired formula.  

3) To justify our formulas we require some transversality conditions.
These conditions are certainly satisfied if the degree (entering as a
parameter to the formulas of previous section) is large enough. The
cases of small degrees require additional investigations similar to
those of~\cite{KP2}. For example, it is useless to count multitangent
lines to a cubic surface since it contains whole lines.

4) Even if the existence of the residue polynomials and the universal
characteristic classes of multisingularities is established there is
no hope to find any close form formula for a series of singularities
(say, for the singularity class $A_r$, $r=1,2,\dots$ or for the class
of multiple points $\om_r$, $r=1,2,\dots$). The reason is that the
complete classification of singularities is highly irregular. The
computation of these classes (either using the intersection theory
methods of blowing-ups, Hilbert schemes, and residual intersections
or more simple method of Rim\'anyi of using symmetries) is based on the
complete classification of singularities which is unknown.

5) Nevertheless, one may hope to obtain the corresponding formulas
for the series of singularity classes if these classes are considered
after certain factorization. It means that the desired formula can be
obtained up to some terms which vanish for the given geometrical
problem. A typical example is the enumeration of rational nodal
curves on surfaces, see the results of
Kontsevich-Manin~\cite{KM} and Bryan-Leung~\cite{BL1,BL2,BL3}
(vanishing higher Chern classes $c_i$, $i>2$, for a surface kills a
large part of the ring of all Legendre characteristic classes).
Results of Kleiman~\cite{Kl2,Kl3}, Katz~\cite{Katz},
Colley~\cite{Co1}, on the maps possessing only singularities of
corank $1$ is another example. The arguments used for finding such
formulas rely very much on the particularity of the problem.
Nevertheless, even in this case the knowledge of the existence of a
universal formula can be useful. 


%\newpage
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\end{document}
