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\begin{document}

\title{ Simple singularities of curves}

\author{ V.I.Arnold
\thanks{Partially supported by the Russian Basic Research Foundation,
project 96-01-01104.The author thanks the Mittag-Leffler Institute
where this paper has been written, for the support, thanks
R.Fr\"oberg and B.Shapiro for suggestions and references and thanks
O.Viro and A.Soloviev for the typing help}\\
{\small Steklov Mathematics Institute,
Moscow}\\
{\small \& CEREMADE, Universit\'e Paris-Dauphine}}

\date{}
\maketitle

\rightline{Dedicated to L.D.Faddeev}

\section{  Introduction } %s1

{\it A singularity of a curve} below means a germ of
 a holomorphic mapping of the
complex line into the complex space at a singular
point (where the derivative of the mapping vanishes) considered up to
the biholomorphic mappings of the image space.

The singularity is called {\it simple} if all neighbouring singularities
belong to a finite set of equivalence classes (have no moduli). The simple
singularities of curves in the plane have been classified by J.W.Bruce
and T.Gaffney \cite{1}, that of curves in three-space -- by C.Gibson and
C.Hobbs \cite{7}.

The singularity is called {\it stably simple} if it is simple and remains
simple
when the ambient space is embedded into a larger space.

Two curves, obtained one from the other by such an embedding, are called
{\it stably equivalent.}

We classify below the simple curve singularities in spaces of any
dimension up to stable equivalence.

We find, that almost all those singularities, whose Tailor series starts
from a term of degree 2 or 3  or has the form
$$
x = t^4,\quad y = t^6,\quad z_i = 0\quad({\rm mod}\ t^7),
$$
are simple, and that besides these curves there exist 32
{\it sporadic simple curves}, listed below.

"Almost all" curves means here and below "all curves except the infinitely -
degenerate curves, forming a codimension infinity set in the space of
all curves".

The {\it codimension}
 here and below means "the codimension in the space
of Taylor  series with zero constant terms."

The codimension of the set of all nonsimple singularities of
 germs of curves belonging to the $N$-space equals $6N-6$. This means,
that the non-simple singularities never occur on curves of typical
$6N-8$ - parametric families of curves, but can't be eliminated by a small
deformation of the family, if the number of parameters reaches $6N-7.$

All simple singularities, whose Taylor series starts from terms
of degree $n,$ are realised in the $n$-space.         From the
32 sporadic singularities one requires the 6-space, four --  the 5-space,
 9 -- the 4-space, 13 -- the 3-space, five -- can be realised in the plane.

>From the point of view of algebra we study the "coartinian" subalgebras
(of finite codimension in the algebra of power series in one variable).
The main tools are {\it the singularity semigroup,} formed by the orders
of the principal terms of the series forming the subalgebra (see \cite{2,3,4})
and {\it the spectral sequence} of a quasihomogeneous filtration, similar to
that used in the theory of critical points of functions (see \cite{5}).

Two natural filtrations in the subalgebra and the set of the
commuting nilpotent matricies representing our algebra
suggest that the classification of the simple (having no continious
invariants or moduli) coartinian subalgebras might be included into some
theory similar to that of quivers.  But up to now I was unable to
find any relation of our classification problem to the known problems of this
type (including the classifications of the simple Lie algebras, of the simple
categories of vector spaces, of the simple caustics and wave fronts,
of the regular polyhedra, of the Coxeter groups, of the simple meromorphic
functions and the others " $A,D,E $ -- classifications").

The special role of the number 6 as of the boundary between the simple
and non simple singularities suggest that the usual $ E_6,\quad E_7,\quad
E_8 $ have something to do with our problem.  This "six" is responsible for
the tangent $7/6$ of the upper boundary of the domain of the "good
dimensions" of John Mather \cite{6}.

I was led to the classification of the simple singularities of curves by
an attempt to construct the local algebraic (or better analytic)
symplectic geometry in terms of the combination of the
Poisson bracket and of the usual multiplication of functions
in the algebra of functions on the phase space. I hope to describe
elsewhere the symplectic corollaries of the results of the
present paper.

\section{The list of the stably simple and of the fencing singularities
of curves}

A monomial curve in $\C^N,$ defined by the parametrical equations
$$
x_1=t^{a_1},\quad\ldots,\quad x_N=t^{a_N}\quad (a_1 < \dots < a_N),
$$
will be denoted by the symbol $(a_1,\ldots,a_N)$. If the monomial
$t^{a_i}$ is replaced by the binomial $t^a+t^b$ in one of the components,
the element $ a_i$ of the symbol will be replaced by $(a,b)$. For
instance, the symbol $(4,(6,7))$  denotes the plane curve
 $x_1=t^4$, $x_2=t^6+t^7$.

\begin{theorem}\label{T1} %1
The stably simple singularities of curves are stably equivalent to the
singularities of the following list (which are all simple):
\begin{enumerate}
\item almost all the curves with a nonzero quadratic part
of the Taylor series;
\item almost all the curves with zero quadratic and nonzero cubic part
of the Taylor series;
\item almost all the curves whose  6-jet is $(t^4,t^6,0,\ldots,0)\,$;
\item thirty two sporadic curves, listed below.
\end{enumerate}
\end{theorem}

\medskip

{\bf Remark.}
The infinitely degenerate curves, eliminated by the words "almost all"
are not embedded topologically (the curve "covers itself", as, for instance,
the curve with the symbol $(2,6)$ ). For the formal series and for
real smooth curves the infinite degeneracy is defined by the same
infinite system of equations on the Taylor coefficients, as in  the holomorphic
case, but the toplogical description is not that simple.

\bigskip

\subsection*{The list of sporadic curves.}
This list contains five parts.

\begin{enumerate}

\item \it{Seven curves whose symbol starts with $(4, 5)$}:
$$
\begin{array}{cccccccc}
(4,5,6,7)^{3N}  \\
\uparrow            \\
(4,5,6)^{4N-3} &\leftarrow&  (4,5,7)^{4N-2}&\leftarrow&
        (4,(5,7),11)^ {5N-4} &\leftarrow& (4,5,11)^{5N-3}             \\
&&&&          \uparrow           &    &       \uparrow                  \\
&&&&   (4,(5,7))^{6N-6}    &\leftarrow&  (4, 5)^{6N-5}                \\
\end{array}
$$


{ \it This part contains all the curves whose symbol starts with
$(4, 5)$.  All curves with this property are simple.

The upper index means the codimension, and arrows  --
the adjunctions.}

\item {\it Fifteen curves whose symbol starts with  $(4,7)$}:
$$
\begin{array}{ccccc}
(4,7,9,10)^{5N-2}       \\
      \uparrow              \\
(4,7,(9,10))^{6N-5}     &\leftarrow&(4,7,9)^{6N-4}    \\
       \uparrow                                     \\
(4,(7,9),10,13)^{6N-4}  &\leftarrow&(4,7,10,13)^{6N-3}        \\
      \uparrow              & &           \uparrow              \\
(4,(7,9),10)^{7N-7}     &\leftarrow&(4,7,10)^{7N-6}           \\
      \uparrow              &    &        \uparrow              \\
(4,(7,9),13)^{7N-6}     &\leftarrow&(4,7,13)^{7N-5}           \\
      \uparrow              &                       \\
(4,(7,9),17)^{8N-8}     &\leftarrow&(4,(7,13),17)^{8N-7} &\leftarrow&(4,7,17)^{8N-6}\\
      \uparrow              &    &        \uparrow         &    &       \uparrow     \\
(4,(7,9))^{9N-10}       &\leftarrow&(4,(7,13))^{9N-9}    &\leftarrow&(4,7)^{9N-8}.\\
\end{array}
$$


This part contains all the curves whose symbol
starts with $(4,7)$. All the curves with this property are simple.

\item \it{The curve $(4,9,10,11)^{6N-3}$}.

\item \it{Six curves whose symbol starts with $(5)$}:
$$
\begin{array}{ccccc}
(5,6,7,8,9)^{4N} &\leftarrow& (5,6,7,8)^{5N-4} &\leftarrow& (5,6,7,9)^{5N-3} \\
                                         &&&&           \uparrow    \\
(5,7,8,9)^{6N-5} &\rightarrow& (5,7,8,9,11)^{5N-1} &\rightarrow& (5,6,8,9)^{5N-2}
\end{array}
$$

\item \it{Three curves whose symbol starts with  $(6)$}:
$$
\begin{array}{ccccc}
(6,7,8,9,10,11)^{5N} &\leftarrow& (6,7,8,9,10)^{6N-5} &\leftarrow& (6,7,8,9,11)^{6N-4}
\end{array}
$$
\end{enumerate}


\begin{theorem} %2
Each nonsimple singularity of curve either belongs to one of the three
infinitely
degenerate limit classes of the families 1,2 and 3 of theorem \ref{T1},
or is adjacent to one of the following three classes of fencing curves:
$$
\begin{array}{c}
(t^4+t^5+at^6, t^9, t^{10}, t^{15}),\\
(t^5, t^6+t^8+at^9,t^7),\\
(t^6,t^7+t^{10}+at^{11}, t^8+bt^{10}, t^9).
\end{array}
$$
The codimensions of these three classes are equal to  $7N-6$,\quad $6N-6$,
\quad
$7N-8$ respectively.
\end{theorem}

For instance, the codimension of the set of nonsimple singularities of germs of
curves in  $\C ^N$ equals $6N-6$, if $N\geq 3$. The coefficient 6 defines
the slope $7/6$ of the upper boundary of the domain of the good
dimensions of J.Mather at infinity.


\section{Classification of curves having nonzero 3-jet} %s3

All such curves, except those which are infinitely degenerate, are
simple and are stably equivalent either to plane curves or
to curves of the 3-space.

\begin{theorem} %3
The singular curves whose 2-jet is nonzero (except the infinitely degenerate ones)
form one infinite series $A_{2k}$ of stable plane curves:
$$
\begin{array}{ccccccccc}
(2,3)^{N}& \leftarrow & (2,5)^{2N-1} & \leftarrow & \dots &\leftarrow &
(2,2k+1)^{kN-(k-1)}& \leftarrow & \dots
\end{array}
$$
\end{theorem}

Suppose now that the quadratic part of the Taylor series vanishes but the
cubic part does not vanish.

\begin{theorem} %4
The singular curves having zero 2-jet and nonzero 3-jet (except those which
are infinitely
degenerate) form two 3-indices series of curves, stably equivalent to
 the following curves in the three--space:
$$
\begin{array}{lcl}
E_{6k,p,i}  &=&(3,(3k+1,3k+2+3i),3k+2+3p)^{(2k+p)N-(2k+2p-2-i)}, \\
E_{6k+2,p,i}&=&(3,(3k+2,3k+4+3i),3k+4+3p)^{(2k+p+1)N-(2k+2p-1-i)},\\
\end{array}
$$
where $k\geq 1$, $0\leq p\leq k$, $0\leq i\leq{k-1}$.
\end{theorem}

{\bf Remark.}
For the boundary values of the indices one can simplify the
normal form, choosing

$$
\begin{array}{lcl}
E_{6k,k,i}    &=&(3,(3k+1,3k+2+3i))^{3kN-(4k-2-i)},     \\
E_{6k+2,k,i}  &=&(3,(3k+2,3k+4+3i))^{(3k+1)N-(4k-1-i)}, \\
E_{6k,p,p}    &=&(3,3k+1,3k+2+3p)^{(2k+p)N-(2k+p-2)},   \\
E_{6k+2,p,p}  &=&(3,3k+2,3k+4+3p)^{(2k+p+1)N-(2k+p-1)}, \\
E_{6k,k,k-1}  &=&(3,3k+1)^{3kN-(3k-1)},                 \\
E_{6k+2,k,k-1}&=&(3,3k+2)^{(3k+1)N-3k}.
\end{array}
$$

\medskip

One has the adjacencies
$$
\begin{array}{ccccc}
E_{.,p,i} &\rightarrow & E_{.,p,i-1} &\rightarrow & E_{.,p-1,i-1}.
\end{array}
$$

\medskip

The hierarchy of these $E$ -- singularities starts with:
$$
\begin{array}{cccccccccc}
(3,4,5)^{2N} &\leftarrow & (3,4)^{3N-2} &\leftarrow & (3,5,7)^{3N-1}
 &\leftarrow & (3,5)^{4N-3} \\
             &    &              &    &     \uparrow        \\
 && && (3,7,8)^{4N-2} &\leftarrow & (3,(7,8),11)^{5N-4} &\leftarrow & \cdots \\
 && &&        \uparrow    &    &      \uparrow \\
 && && (3,8,10)^{5N-3} &&   (3,7,11)^{5N-3} &\leftarrow & \cdots \\
 && &&        \uparrow \\
 && &&        \vdots
\end{array}
$$

\section{Classification of curves having the  6-jet $(t^4,t^6,\dots)$} %s4

All such curves, except those which are infinitely degenerate, are simple.
\begin{theorem} %5
Curves having the 6-jet $(t^4, t^6, \ldots),$ which are not infinitely
degenerate, form seven one-indice families:
$$
\begin{array}{rll}
a_k=&\!\!(4,(6,2k+1))^{(5+k)N-(6+k)},\hfill             &k\geq3;     \\   \\
b_k=&\!\!(4,(6,2k+3),2k+11)^{(5+k)N-(5+k)},             &k\geq2;        \\  \\
c_k=&\!\!(4,(6,2k+5),2k+9)^{(5+k)N-(4+k)},              &k\geq1;        \\    \\
d_k=&\!\!(4,(6,2k+7),2k+9, 2k+11)^{(5+k)N-(3+k)},       &k\geq0;        \\
\\
  e_k=&\!\!(4,(6,2k+9),2k+11)^{(5+k)N-(2+k)},               &k\geq{-1};     \\
\\
f_k=&\!\!(4,6,2k+5,2k+7)^{(5+k)N-(5+k)},                &k\geq1;        \\ \\
g_k=&\!\!(4,6,2k+7)^{(5+k)N-(4+k)},                     &k\geq0.
\end{array}
$$

These seven series adjacencies are:
$$
\begin{array}{ccccccccc}
%[height=1.7em]
b_{k-1} &\lTo& c_{k-1} &\lTo& d_{k-1} &\lTo& e_{k-1}            \\
        &    &         &    & \uTo    &    & \uTo               \\
\uTo    &    & \uTo        &    & f_k     &\lTo&  g_k               \\
        &    &     &    & \uTo    &    & \uTo               \\
a_k     &\lTo& b_k     &\lTo& c_k     &\lTo& d_k    &\lTo& e_k  \\
        &    & \uTo    &    & \uTo    &    & \uTo   &    & \uTo \\
        &    & \vdots  &    & \vdots  &    & \vdots &    & \vdots
\end{array}
$$
\end{theorem}




The hierarchy of these singularities starts with
$$
\begin{array}{ccccccccccc}
%[height=1.7em]
e_{-1}                                                          \\
\uTo                                                            \\
g_0                                                             \\
\uTo                                                            \\
d_0 &\lTo& e_0                                                  \\
\uTo&    & \uTo                                                 \\
f_1 &\lTo& g_1                                                  \\
\uTo&    & \uTo                                                 \\
c_1 &\lTo& d_1 &\lTo& e_1                                       \\
    &    & \uTo&    & \uTo                                      \\
\uTo&    & f_2 &\lTo& g_2                                       \\
    &    & \uTo&    & \uTo                                      \\
b_2 &\lTo& c_2 &\lTo& d_2 &\lTo& e_2                            \\
    &    &     &    & \uTo&    & \uTo                           \\
\uTo&    & \uTo&    & f_3 &\lTo& g_3                            \\
    &    &     &    & \uTo&    & \uTo                           \\
a_3 &\lTo& b_3 &\lTo& c_3 &\lTo& d_3 &\lTo& e_3                 \\
    &    &     &    &     &    & \uTo&    & \uTo                \\
    &    & \uTo&    & \uTo&    & f_4 &\lTo& g_4                 \\
    &    &     &    &     &    & \uTo&    & \uTo                \\
    &    & a_4 &\lTo& b_4 &\lTo& c_4 &\lTo& d_4 &\lTo& e_4      \\
    &    &     &    & \uTo&    & \uTo&    & \uTo&    & \uTo     \\
    &     &    &    &\vdots&   &\vdots&   &\vdots&   &\vdots    \\
\end{array}
$$

\section{The coartinian algebra and the semigroup of
the curve singularity.} %s5


The classification of the curve singularities can be interpreted in dual terms
as the listing of "coartinian" subalgebras of finite codimension
in the algebra of power series in one variable
(considering the subalgebras up to the automorphisms of the ambient
algebra of the series, that is up to the formal diffeomorphisms of the line).

\medskip
{\bf Definition.}
{\it The local algebra}  $A_f$ of the curve $x_i=f_i(t)$
in $\C ^N$ is the subalgebra of the algebra of the power series in $t$,
generated (as the closure with respect to the powers of the maximal
ideal) by the multiplicative generators $f_i.$  In other terms it is
the algebra
$$
A_f = f^* {\cal M}  \subset \C[[t]],
$$

where ${\cal M}$ is the maximal ideal of the ring $ \C[[x_1, \ldots, x_N]].$

\medskip

The filtrations of the algebras of the series in $t$ and in $x$
according to the order of the zero at the origin generate in the algebra
$A_f$ two filtrations. Denote by $M^i$ the ideal of the algebra
$\C [[t]]$, formed by the series, divisible by  $t^i$. The filtration
$$
A_f={A_f}^0 \supset{A_f}^1 \supset\cdots,\quad {A_f}^s=A_f \cap M^s,
$$
coincides with $M^i \supset M^{i+1}\supset\cdots$ from some place
 $i$,
since the codimension of $A_f$ in $\C[[t]]$ is finite (provided that
the singularity is not infinitely degenerate).

We obtain a finite-dimensional commutative algebra $A_f/M^i$ (equiped
with a filtration), represented by commuting nilpotent linear
operators acting in the finite-dimensional vector-space $M^1/A_f$.

The dimension $a=\dim_\C M^1/A_f$ of this vector space appears in the
 expression  $aN-b$ for the codimension of the class of the simple singularity
  $f$ of a curve belonging to $\C^N$.
The correction $b$ equals to the dimension of the quotient space
$(M^1df/dt + {A_f}^N)/{A_f}^N$, as it follows from the formula for the
 tangent space to the orbit of the germ $f$.

\medskip
{\bf Definition.}
{\it The semigroup} $P_f$ of the curve singularity $x_i=f_i(t)$ is the
 subsemigroup of the additive semigroup of natural numbers formed by
 the orders of zero at the origin of all the linear
 combinations of the products of the series
$f_i$.

\medskip

  In other terms, the semigroup $P_f$ consists of the degrees of the one
(nonzero) dimensional homogeneous components of the graded algebra
$G_f$, associated with the filtred algebra
$A_f \supset {A_f}^1 \supset\dots$ :
$$
{G_f}^i = {A_f}^i / {A_f}^{i+1},\quad G_f = {\textstyle\bigoplus}
{G_f}^i.
$$

Using this terminology we can say that {\it the graded algebra
$G_f$, corresponding to the curve $f$, is the semigroup algebra of the
semigroup  $P_f$ of this curve} (the semigroup algebra consists
of the linear combinations of the monomials $t^i$, whose exponents $i$
belong to the semigroup).

\medskip
{\bf Example.}
The semigroup of the plane curve germ $x=t^3$, $y= \sin^3 t$ is
 described by the line
$$
 P_f = ([1],[2],{\bf 3},[4],{\bf 5},6,[7],8.).
 $$
Here the square brackets enclose the {\it gaps} $[i]$, (for which
$\dim {G_f}^i = 0$ ), and the bold entries represent the degrees of
the multiplicative generators of the algebra $G_f$.
The point following $8$ means that all the larger integers
$(9, 10, \ldots)$ do belong to the semigroup.

\medskip

The hierarchy of the singularities, described in our theorems,
reflects the semigroups arithmetics.
 For the semigroups generated by two elements one has
 a very useful {\it Sylvester duality} between the semigroup
 and its complement. Let $u$  and $v$  be two relatively prime natural
 numbers. Denote by $K$ the even number $(u-1)(v-1)$.

Sylvester ~\cite{3} has published in a nonmathematical journal the
following

{\bf Proposition.}
{\it An integer $c$ is representable as a linear combination
 $mu+nv$ with integer nonnegative coefficients if and only if the number
  $K-1-c$, (symmetric to $c$ with respect to the middle of the segment
   $[0, K-1]$), is not representable in this form.}

\medskip

For instance, all the integers greater or equal to $K$ are
representable, while   $K-1$ is not representable in the form
$mu+nv$, $m\ge0$, $n\ge0$.
The representable numbers form exactly one half of the segment  $[0,K-1].$

\medskip

{\bf Example.}
For $u=3$, $v=5$ the semigroup of combinations is
$$
(0, [1], [2], {\bf 3}, [4], {\bf 5},  6, [7], 8. ).
$$
The sequence of the lengths of the gaps $(2,1,1)$  is dual to
the sequence $(1,1,2)$  of the lengths of the antigaps.
Among the eight integral points of the segment
$[0,7]$  four do and four do not belong to the semigroup
of the representable points.

\medskip

\section{ Technics of calculations using semigroups} %s6

The proofs of the classification theorems formulated above
consist of long calculations, which can be made rather short
in terms of the semigroups. I shall show how the
semigroup technics works in a particular example proving the
sufficiency of the symbol $(3,7,8)$ providing a polynomial normal form
for any curve  $(t^3+\cdots, t^7+\cdots, t^8+\cdots).$ We shall kill
all the higher order terms by the consequtive changes of the independent and
of the dependent variables.


The change of the dependent variables  $x_i$ is called {\it  left},
the change of the independent variable $t$ is called the {\it right}
change of variables.

The right change  $t=T(1+aT^r)$  transforms the monomial $t^m$
to its sum with the increment of the form
$t^m \sum c_k t^{kr},\penalty0 \quad k\geq 1$.

The left change  $X_i=x_i+g(x)$ adds to $x_i$ a
correction series, whose support belongs to the semigroup
 $P_f$. Choosing  $g$ one can obtain any initial term  $t^j$ in the
 correction series, provided that $j$ belongs to the semigroup.
 One should only control, that the correction would
 provide a genuine change of variables, verifying the
 implicite function theorem condition. It suffices that $j$ be larger
 than the lowest term of the series  $f_i.$


We shall denote the series like $t^3+at^5+o(t^5)$  by the symbol like
$(3\amp 5.).$ The dot inside the brackets is important -- it
means that the addition of the series  $o(t^5)$ is permitted.

{\bf Proposition.}
{\it Every curve  $(3.,7.,8.)$  is equivalent to the monomial curve $(3,7,8)$.
}

\medskip
{\bf Proof.}
The semigroup has the description
$$
([1],[2],3,[4],[5],6.).
$$

Killing one by one all the summands whose degrees belong to
the semigroup in each component, we get the left
equivalence
$$
(3.,7.,8.) \sim (3\amp 4\amp 5,7,8)
$$
of the initial curve to the curve, each component of which has as its
support the union of the gaps of the semigroup which follow
the exponent of the leading term of this component (thus, the gaps
[4] and [5] follow the exponent 3 of the leading term of the first
component).


Now we kill the  $t^4$  in the first component using the right equivalence
with  $r=1$. Next we kill by the left equivalence the newborn
increment terms (of degree 6 or larger in the first, 8 or larger in the
second, 9 or larger in the third component).
We thus get the equivalence
$$
    (3\amp 4\amp 5,7,8) \sim (3\amp 5,7,8).
$$

The remaining perturbation $t^5$  in the first component can be killed
by the right equivalence with $r=2$. The newborn increment terms have
degrees  7 or higher in the first, 9 or higher in the second, 10 or
higher in the third component. All their exponents  do belong to the
semigroup. Therefore all these newborn terms can be killed by
a left change of variables :
$$
     (3\amp 5,7,8) \sim (3,7,8).
$$

Thus we get the required equivalence to the normal form.

\medskip

{\bf Remark.}
Strictly speaking, all these blood-thirsty reasonings are valid only at
the formal series level. However in the case of
the holomorphic (or even real smooth) curves the changes of variables might
be made holomorphic (or smooth, respectively), as it is well
known
from the
Weierstrass--Malgrange preparation theorem of singularity theory.


\medskip

The proofs of the remaining theorems follow the same pattern,
but the calculation of the semigroup is sometimes less obvious
(the semigroup of a curve is {\bf not} defined by the exponents of
the leading terms of the components).
For instance, the semigroup of the curve
defined by the symbol  $(4,(6,7+2k))$ contains  $13+2k$.

To study the curves whose 6-jet is $(t^4,t^6,0,\ldots,0)$
one should use the stationary group of the plane curve
$x^3=y^2$.  The Lie algebra of this group is generated ( as a module
over the algebra of functions) by the Euler quasihomogeneous field
 $2x\partial_x+3y\partial_y$  and by the Hamiltonian field
$2y\partial_x+3x^2\partial_y.$

The spectral sequence of the corresponding quasihomogeneous filtration provides
immediately the equivalence
$$
(t^4, t^6+t^{7+2k}+o(t^{7+2k})) \sim (t^4, t^6+t^{7+2k}),
$$
and reduces to the normal forms (described in theorem 5) the
other components of the corresponding space curve. Thus one gets,
for instance, the nonimmediate equivalence
$$
(t^4,t^6+t^{7+2k},t^{9+2k}+at^{11+2k}) \sim
(t^4,t^6+t^{7+2k},t^{9+2k}).
$$



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\end{document}

