Moduli Spaces of Germs of Semiquasihomogeneous Legendrian Curves

TL;DR

This paper constructs a moduli space for Legendrian curve singularities, focusing on those that are conormals of plane curves with a single Puiseux pair, using a contact analogue of the Kodaira-Spencer map.

Contribution

It introduces a new moduli space framework for Legendrian curve singularities, specifically for conormal curves with one Puiseux pair, via a contact Kodaira-Spencer map.

Findings

Established a contact analogue of the Kodaira-Spencer map for Legendrian curves.

Constructed a moduli space classifying Legendrian curve singularities.

Focused on conormal curves of plane curves with one Puiseux pair.

Abstract

We construct a moduli space for Legendrian curves singularities which are contactomorphic-equivalent and equisingular through a contact analogue of the Kodaira-Spencer map for curve singularities. We focus on the specific case of Legendrian curves which are the conormal of a plane curve with one Puiseux pair.

Equations

$${\hat{\rho }}_{\phi }:{\phi }^{\ast }{\Omega }_{X_2/S}^1={\mathcal{O}}_{X_1}{\otimes }_{{\phi }^{-1}{\mathcal{O}}_{X_2}}{\phi }^{-1}{\Omega }_{X_2/S}^1\to {\Omega }_{X_1/S}^1.$$

$${\rho }_{\phi }:X_1{\times }_{X_2}{T}^{\ast }(X_2/S)\to T^{\ast }(X_1/S).$$

$$V_i=\{(x,\xi )\in V:{\xi }_i\ne 0\},{\omega }_i={\xi }_i^{-1}\theta ,i=1,...,n.$$

$${\mathbb{P}}_{\mathcal{Y}}^{\ast }(M|S)={\cup }_{i=1}^r{\mathbb{P}}_{{\mathcal{Y}}_i}^{\ast }(M|S).$$

$$\mathcal{L}\hookrightarrow {\mathbb{C}}^3\times S\to S$$

$$\chi \circ i=i,q\circ \chi =q\mathrm{and}\chi (0,s)=(0,s)for\; eachs.$$

$$(\ell +1)p^{\ell }{f}_x+\ell p^{\ell +1}{f}_y\equiv h_{\ell }mod{I}_L.$$

$$G(x,y,s_1,...,s_n)=f(x,y)+\sum _{i=1}^n{s}_i{g}_i$$

$$H^{\gamma }\equiv p^{\gamma }{\partial }_{x}Fmod{I}_{Con(\mathcal{F})}+{\Delta }_F.$$

$$H^{\gamma }\equiv (\gamma +1)p^{\gamma }{\partial }_{x}F+\gamma p^{\gamma +1}{\partial }_{y}Fmod{I}_{Con(\mathcal{F})}.$$

$${\partial }_{x}F+p{\partial }_{y}F\equiv 0mod{I}_{Con(\mathcal{F})}.$$

$${\partial }_{x}F{\partial }_{t}X+{\partial }_{y}F{\partial }_{t}Y=0.$$

$$\chi \circ i=i,q\circ \chi =q\mathrm{and}\chi (0,s)=(0,s)for\; eachs.$$

$$\chi (x,y,p,s)=(x+\alpha ,y+\beta ,p+\gamma ,s).$$

$$\left(1+\frac{\partial \alpha }{\partial x}+p\frac{\partial \alpha }{\partial y}\right)\frac{\partial \beta }{\partial p}-p\frac{\partial \alpha }{\partial p}\frac{\partial \beta }{\partial y}-\frac{\partial \alpha }{\partial p}\frac{\partial \beta }{\partial x}=p\frac{\partial \alpha }{\partial p},$$

$$\frac{\partial \beta }{\partial p}=p\frac{\partial \alpha }{\partial p},\beta +p{\mathcal{O}}_{X\times T_{\epsilon }}={\beta }_0.$$

$$\Phi (x,y,p)=(\lambda x,\lambda \mu y,\mu p),\lambda ,\mu \in {\mathbb{C}}^{\ast },$$

$$\Phi (x,y,p)=(ax+bp,y+\frac{ac}{2}{x}^2+\frac{bd}{2}{p}^2+bcxp,cx+dp),\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=1,$$

$$\alpha ,\beta ,\gamma ,{\beta }_0,\frac{\partial \alpha }{\partial x},\frac{\partial {\beta }_0}{\partial x},\frac{\partial \beta }{\partial p},\frac{{\partial }^2\beta }{\partial x\partial p}\in (x,y,p).$$

$$Y\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\hat{\tau }(f):=di{m}_{\mathbb{C}}\frac{\mathbb{C}\{x,y\}}{I_f^{\mu }}$$

$$B$$

$$C$$

$$D$$

$$A_0$$

$$F_A=f+\sum _{\ell \in I_A}{s}_{o(\ell )}{m}_{\ell }$$

$$X(t,s_A)=t^k,Y(t,s_A)=\sum _{i\ge 0}{\psi }_i{t}^{n+i}$$

$$a\cdot \Phi (t,s_A)=\left(a^k{t}^k,a^n\left(\theta t^n+\sum _{i\ge 1}(a\cdot {\psi }_i)a^i{t}^{n+i}\right)\right).$$

$$t\mapsto {\Phi }_a(t,s_A)=\left(t^k,\theta t^n+\sum _{i\ge 1}(a\cdot {\psi }_i)a^i{t}^{n+i}\right)$$

$${\Psi }^{\ast }(p^{\gamma }{\partial }_x{F}_A-H_A^{\gamma })\in t^c\mathbb{C}[s_A]\{t\}.$$

$$p^{\gamma }{\partial }_x{F}_A-H_A^{\gamma }\in I_{Con({\mathcal{F}}_A)}+{\Delta }_{F_A}.$$

$${\Theta }_A\hookleftarrow {\Theta }_{A,A^{\prime }}\stackrel{r_{A,A^{\prime }}}{⟶}{\Theta }_{A^{\prime }},$$

Full text

1. Introduction

Greuel, Laudal, Pfister et all (see [5], [8]) constructed moduli spaces of germs of plane curves equisingular to a plane curve $\{y^{k}+x^{n}=0\}$, $(k,n)=1$. Their main tools are the Kodaira Spencer map of the equisingular semiuniversal deformation of the curve and the results of [6]. We extend their results to Legendrian curves.

Let $Y$ be the germ of a plane curve that is a generic plane projection of a Legendrian curve $L$. The equisingularity type of $Y$ does not depend on the projection (see [12]). Two Legendrian curves are equisingular if their generic plane projections are equisingular. We say that an irreducible Legendrian curve $L$ is semiquasihomogeneous if its generic plane projection is equisingular to a quasihomogeneous plane curve $\{y^{k}+n^{n}=0\}$, for some $k,n$ such that $(k,n)=1$. Hence the generic plane projection of $L$ is a semiquasihomogeneous plane curve.

In section 2 we recall the main results of relative contact geometry. In section 3 we construct the microlocal Kodaira Spencer map and study its kernel $\mathcal{L}_{B}$, a Lie algebra of vector fields over the base space $\mathbb{C}^{B}$ of the semiuniversal equisingular deformation of the plane curve $\{y^{k}+n^{n}=0\}$. We use $\mathcal{L}_{B}$ in order to construct a Lie algebra of vector fields $\mathcal{L}_{C}$ over the base space $\mathbb{C}^{C}$ of the microlocal semiuniversal equisingular deformation of $\{y^{k}+n^{n}=0\}$. In section 4 we recall some results of [6]. In section 5 we study the stratification of $\mathbb{C}^{C}$ induced by $\mathcal{L}_{C}$ and show that the conormals of two fibers $\mathcal{F}_{b}$, $\mathcal{F}_{c}$ of the microlocal semiuniversal equisingular deformation of $\{y^{k}+n^{n}=0\}$ are isomorphic if and only if $b$ and $c$ are in the same integral manifold of $\mathcal{L}_{C}$. Moreover, we construct the moduli spaces. The final section in dedicated to presenting an example.

2. Relative contact geometry

Let $q:X\to S$ be a morphism of complex spaces. We can associate to $q$ a coherent $\mathcal{O}_{X}$-module $\Omega^{1}_{X/S}$, the sheaf of relative differential forms of $X\to S$, and a differential morphism $d:\mathcal{O}_{X}\to\Omega^{1}_{X/S}$ (see [7] or [9]).

If $\Omega^{1}_{X/S}$ is a locally free $\mathcal{O}_{X}$-module, we denote by $\pi=\pi_{X/S}:T^{*}(X/S)\to X$ the vector bundle with sheaf of sections $\Omega^{1}_{X/S}$. We say that $T(X/S)$ [$T^{*}(X/S)$] is the relative tangent bundle [cotangent bundle] of $X\to S$.

Let $\varphi:X_{1}\to X_{2}$, $q_{i}:X_{i}\to S$ be morphisms of complex spaces such that $q_{2}\varphi=q_{1}$. There is a morphism of $\mathcal{O}_{X_{1}}$-modules

[TABLE]

If $\Omega^{1}_{X_{i}/S}$, $i=1,2$, and the kernel and cokernel of (2.1) are locally free, we have a morphism of vector bundles

[TABLE]

If $\varphi$ is an inclusion map, we say that the kernel of (2.2), and its projectivization, are the *conormal bundle of $X_{1}$ relative to $S$. *We will denote by $T^{*}_{X_{1}}(X_{2}/S)$ or $\mathbb{P}^{*}_{X_{1}}(X_{2}/S)$ the conormal bundle of $X_{1}$ relative to $S$.

Assume $M$ is a manifold. When $q$ is the projection $M\times S\to S$ we will replace ”$M\times S/S$” by $"M|S"$. Let $r$ be the projection $M\times S\to M$. Notice that $\Omega^{1}_{M|S}\xrightarrow{\sim}\mathcal{O}_{M\times S}\otimes_{r^{-1}\mathcal{O}_{M}}r^{-1}\Omega^{1}_{M}$ is a locally free $\mathcal{O}_{M\times S}$-module. Moreover, $T^{*}(M|S)=T^{*}M\times S$.

We say that $\Omega^{1}_{M|S}$ is the *sheaf of relative differential forms of $M$ over $S$. *We say that $T^{*}(M|S)$ is the relative cotangent bundle of $M$ over $S$.

Let $N$ be a complex manifold of dimension $2n-1$. Let $S$ be a complex space. We say that a section $\omega$ of $\Omega^{1}_{N|S}$ is a relative contact form of $N$ over $S$ if $\omega\wedge d\omega^{n-1}$ is a local generator of $\Omega^{2n-1}_{N|S}$. Let $\mathfrak{C}$ be a locally free subsheaf of $\Omega^{1}_{N|S}$. We say that $\mathfrak{C}$ is a structure of relative contact manifold on $N$ over $S$ if $\mathfrak{C}$ is locally generated by a relative contact form of $N$ over $S$. We say that $(N\times S,\mathfrak{C})$ is a *relative contact manifold over $S$. *When $S$ is a point we obtain the usual notion of contact manifold.

Let $(N_{1}\times S,\mathfrak{C}_{1})$, $(N_{2}\times S,\mathfrak{C}_{2})$ be relative contact manifolds over $S$. Let $\chi$ be a morphism from $N_{1}\times S$ into $N_{2}\times S$ such that $q_{N_{2}}\circ\chi=q_{N_{1}}$. We say that $\chi$ is a *relative contact transformation *of $(N_{1}\times S,\mathfrak{C}_{1})$ into $(N_{2}\times S,\mathfrak{C}_{2})$ if the pull-back by $\chi$ of each local generator of $\mathfrak{C}_{2}$ is a local generator of $\mathfrak{C}_{1}$.

We say that the projectivization $\pi_{X/S}:\mathbb{P}^{*}(X/S)\to X$ of the vector bundle $T^{*}(X/S)$ is the *projective cotangent bundle *of $X\to S$.

Let $(x_{1},...,x_{n})$ be a partial system of local coordinates on an open set $U$ of $X$. Let $(x_{1},...,x_{n},\xi_{1},...,\xi_{n})$ be the associated partial system of symplectic coordinates of $T^{*}(X/S)$ on $V=\pi^{-1}(U)$. Set $p_{i,j}=\xi_{i}\xi_{j}^{-1}$, $i\not=j$,

[TABLE]

each $\omega_{i}$ defines a relative contact form $dx_{j}-\sum_{i\not=j}p_{i,j}dx_{i}$ on $\mathbb{P}^{*}(X/S)$, endowing $\mathbb{P}^{*}(X/S)$ with a structure of relative contact manifold over $S$.

Let $\omega$ be a germ at $(x,o)$ of a relative contact form of $\mathfrak{C}$. A lifting $\widetilde{\omega}$ of $\omega$ defines a germ $\widetilde{\mathfrak{C}}$ of a relative contact structure of $N\times T_{o}S\to T_{o}S$. Moreover, $\widetilde{\mathfrak{C}}$ is a lifting of the germ at $o$ of ${\mathfrak{C}}$.

Let $(N\times S,\mathfrak{C})$ be a relative contact manifold over a complex manifold $S$. Assume $N$ has dimension $2n-1$ and $S$ has dimension $\ell$. Let $\mathcal{L}$ be a reduced analytic set of $N\times S$ of pure dimension $n+\ell-1$. We say that $\mathcal{L}$ is a *relative Legendrian variety *of $N\times S$ over $S$ if for each section $\omega$ of $\mathfrak{C}$, $\omega$ vanishes on the regular part of $\mathcal{L}$. When $S$ is a point, we say that $\mathcal{L}$ is a *Legendrian variety *of $N$.

Let $\mathcal{L}$ be an analytic set of $N\times S$. Let $(x,o)\in\mathcal{L}$. Assume $S$ is an irreducible germ of a complex space at $o$. We say that $\mathcal{L}$ is a relative Legendrian variety of $N$ over $S$ at $(x,o)$ if there is a relative Legendrian variety $\widetilde{\mathcal{L}}$ of $(N,x)$ over $(T_{o}S,0)$ that is a lifting of the germ of $\mathcal{L}$ at $(x,o)$. Assume $S$ is a germ of a complex space at $o$ with irreducible components $S_{i},i\in I$. We say that $\mathcal{L}$ is a relative Legendrian variety of $N$ over $S$ at $(x,o)$ if $S_{i}\times_{S}\mathcal{L}$ is a relative Legendrian variety of $S_{i}\times_{S}N$ over $S_{i}$ at $(x,o)$, for each $i\in I$.

We say that $\mathcal{L}$ is a *relative Legendrian variety of $N\times S$ *if $\mathcal{L}$ is a relative Legendrian variety of $N\times S$ at $(x,o)$ for each $(x,o)\in\mathcal{L}$.

Let $Y$ be a reduced analytic set of $M$. Let $\mathcal{Y}$ be a flat deformation of $Y$ over $S$. Set $X=M\times S\setminus\mathcal{Y}_{\rm sing}$. We say that the Zariski closure of $\mathbb{P}^{*}_{\mathcal{Y}_{\rm reg}}(X/S)$ in $\mathbb{P}^{*}(M|S)$ is the conormal $\mathbb{P}^{*}_{\mathcal{Y}}(M|S)$ of $\mathcal{Y}$ over $S$.

Theorem 2.1**.**

The conormal of $\mathcal{Y}$ over $S$ is a relative Legendrian variety of $\mathbb{P}^{*}(M|S)$. If $\mathcal{Y}$ has irreducible components $\mathcal{Y}_{1},...,\mathcal{Y}_{r}$,

[TABLE]

Theorem 2.2**.**

Let $\mathcal{L}$ be an irreducible germ of a relative Legendrian analytic set of $\mathbb{P}^{*}(M|S)$. If the analytic set $\pi(\mathcal{L})$ is a flat deformation over $S$ of an analytic set of $M$, $\mathcal{L}=\mathbb{P}^{*}_{\pi(\mathcal{L})}(M|S)$.

Let $\theta=\xi dx+\eta dy$ be the canonical $1$-form of $T^{\ast}\mathbb{C}^{2}=\mathbb{C}^{2}\times\mathbb{C}^{2}$. Hence $\pi=\pi_{\mathbb{C}^{2}}:\mathbb{P}^{\ast}\mathbb{C}^{2}=\mathbb{C}^{2}\times\mathbb{P}^{1}\to\mathbb{C}^{2}$ is given by $\pi(x,y;\xi:\eta)=(x,y)$. Let $U\,[V]$ be the open subset of $\mathbb{P}^{\ast}\mathbb{C}^{2}$ defined by $\eta\neq 0\,[\xi\neq 0]$. Then $\theta/\eta\,[\theta/\xi]$ defines a contact form $dy-pdx\,[dx-qdy]$ on $U\,[V]$, where $p=-\xi/\eta\,[q=-\eta/\xi]$. Moreover, $dy-pdx$ and $dx-qdy$ define the structure of contact manifold on $\mathbb{P}^{\ast}\mathbb{C}^{2}$.

If $L$ is a germ of a Legendrian curve of $\mathbb{P}^{*}M$ and $L$ is not a fiber of $\pi_{M}$, $\pi_{M}(L)$ is a germ of plane curve with irreducible tangent cone and $L=\mathbb{P}^{*}_{\pi_{M}(L)}M$.

Let $Y$ be the germ of a plane curve with irreducible tangent cone at a point $o$ of a surface $M$. Let $L$ be the conormal of $Y$. Let $\sigma$ be the only point of $L$ such that $\pi_{M}(\sigma)=o$. Let $k$ be the multiplicity of $Y$. Let $f$ be a defining function of $Y$. In this situation we will always choose a system of local coordinates $(x,y)$ of $M$ such that the tangent cone $C(Y)$ of $Y$ equals $\{y=0\}$.

Lemma 2.3**.**

The following statements are equivalent$:$

1. (1)

mult${}_{\sigma}(L)=$mult${}_{o}(Y)$; 2. (2)

$C_{\sigma}(L)\not\supset(D\pi(\sigma))^{-1}(0,0)$; 3. (3)

$f\in(x^{2},y)^{k}$; 4. (4)

if $t\mapsto(x(t),y(t))$ parametrizes a branch of $Y$, $x^{2}$ divides $y$.

Definition 2.4**.**

Let $S$ be a reduced complex space. Let $Y$ be a reduced plane curve. Let $\mathcal{Y}$ be a deformation of $Y$ over $S$. We say that $\mathcal{Y}$ is generic if its fibers are generic. If $S$ is a non reduced complex space we say that $\mathcal{Y}$ is generic if $\mathcal{Y}$ admits a generic lifting.

Given a flat deformation $\mathcal{Y}$ of a plane curve $Y$ over a complex space $S$ we will denote $\mathbb{P}_{\mathcal{Y}}^{\ast}(\mathbb{C}^{2}|S)$ by $\mathcal{C}on(\mathcal{Y})$.

Theorem 2.5** (Theorem $1.3$, [2]).**

Let $\chi:(\mathbb{C}^{3},0)\to(\mathbb{C}^{3},0)$ be a germ of a contact transformation. Let $L$ be a germ of a Legendrian curve of $\mathbb{C}^{3}$ at the origin. If $L$ and $\chi(L)$ are in generic position, $\pi(L)$ and $\pi(\chi(L))$ are equisingular.

Definition 2.6**.**

Two Legendrian curves are equisingular if their generic plane projections are equisingular.

Lemma 2.7**.**

Assume $Y$ is a generic plane curve and $Y\hookrightarrow\mathcal{Y}$ defines an equisingular deformation of $Y$ with trivial normal cone along its trivial section. Then $\mathcal{Y}$ is generic.

Definition 2.8**.**

Let $L$ be (a germ of) a Legendrian curve of $\mathbb{C}^{3}$ in generic position. Let $\mathcal{L}$ be a relative Legendrian curve over (a germ of) a complex space $S$ at $o$. We say that an immersion $i:L\hookrightarrow\mathcal{L}$ defines a deformation

[TABLE]

of the Legendrian curve $L$ over $S$ if $i$ induces an isomorphism of $L$ onto $\mathcal{L}_{o}$ and there is a generic deformation $\mathcal{Y}$ of a plane curve $Y$ over $S$ such that $\chi(\mathcal{L})$ is isomorphic to $\mathcal{C}on\mathcal{Y}$ by a relative contact transformation verifying (2.6).

We say that the deformation (2.3) is *equisingular *if $\mathcal{Y}$ is equisingular. We denote by $\widehat{\mathcal{D}ef}^{es}_{L}$ the category of equisingular deformations of $L$.

Remark 2.9*.*

We do not demand the flatness of the morphism (2.3).

Lemma 2.10**.**

Using the notations of definition 2.8, given a section $\sigma:S\to\mathcal{L}$ of $\mathbb{C}^{3}\times S\to S$, there is a relative contact transformation $\chi$ such that $\chi\circ\sigma$ is trivial. Hence $\mathcal{L}$ is isomorphic to a deformation with trivial section.

Consider the maps $i:X\hookrightarrow X\times S$ and $q:X\times S\to S$.

Theorem 2.11**.**

Assume $\mathcal{Y}$ defines an equisingular deformation of a generic plane curve $Y$ with trivial normal cone along its trivial section. Let $\chi:X\times S\to X\times S$ be a relative contact transformation verifying

[TABLE]

Then $\mathcal{Y}^{\chi}=\pi\left(\chi(\mathcal{C}on\mathcal{Y})\right)$ is a generic equisingular deformation of $Y$.

Definition 2.12**.**

Let $\mathcal{D}ef^{\,es,\mu}_{f}$ (or $\mathcal{D}ef^{\,es,\mu}_{Y}$) be the category given in the following way: the objects of $\mathcal{D}ef^{\,es,\mu}_{f}$ are the objects of $\overset{\twoheadrightarrow}{\mathcal{D}ef^{\,es}_{f}}$; two objects $\mathcal{Y},\mathcal{Z}$ of $\mathcal{D}ef^{\,es,\mu}_{f}(T)$ are isomorphic if there is a relative contact transformation $\chi$ over $T$ such that $\mathcal{Z}=\mathcal{Y}^{\chi}$.

Lemma 2.13**.**

Assume $f\in\mathbb{C}\{x,y\}$ is the defining function of a generic plane curve $Y$. Let $L$ be the conormal of $Y$. For each $\ell\geq 1$ there is $h_{\ell}\in\mathbb{C}\{x,y\}$ such that

[TABLE]

Moreover, $h_{\ell}$ is unique modulo $I_{Y}$.

Definition 2.14**.**

Let $f$ be a generic plane curve with tangent cone $\{y=0\}$. We will denote by $I_{f}$ the ideal of $\mathbb{C}\{x,y\}$ generated by the functions $g$ such that $f+\varepsilon g$ is equisingular over $T_{\varepsilon}$ and has trivial normal cone along its trivial section. We call $I_{f}$ the equisingularity ideal of $f$.

We will denote by $I^{\mu}_{f}$ the ideal of $\mathbb{C}\{x,y\}$ generated by $f,(x,y)f_{x}$, $(x^{2},y)f_{y}$ and $h_{\ell}$, $\ell\geq 1$.

Theorem 2.15** ([9]).**

Assume $Y$ is a generic plane curve with conormal $L$, defined by a power series $f$. Assume $f$ is SQH or $f$ is NND. If $g_{1},...,g_{n}\in I_{f}$ represent a basis of $I_{f}/I^{\mu}_{f}$ with Newton order $\geq 1$ , the deformation $\mathcal{G}$ defined by

[TABLE]

is a semiuniversal deformation of $f$ in $\mathcal{D}ef^{es,\mu}_{f}$.

Lemma 2.16**.**

Let $S$ be the germ of a complex space. Assume $F$ defines an object $\mathcal{F}$ in $\overset{\twoheadrightarrow}{\mathcal{D}ef^{\,es}_{f}}(S)$. Given $\gamma\geq 1$ there are $H^{\gamma}\in\mathcal{O}_{S}\{x,y\}$ such that

[TABLE]

If $f$ has multiplicity $k$, $H^{\gamma}\equiv 0$ for $\gamma\geq k-1$.

Proof.

Let us first show that

[TABLE]

This is a relative version of Lemma $7.2$ of [9]. Since $\mathcal{F}$ is equisingular, the multiplicity and the conductor are constant. Moreover, there are parametrizations of each component of $\mathcal{F}$. Therefore, we can generalize the argument in the proof of the quoted Lemma.

Now it is enough to show that

[TABLE]

Assume $\mathcal{F}$ is irreducible. Let $(t,s)\mapsto(X,Y,P)$ be a parametrization of $Con(\mathcal{F})$. Since $F(X,Y)=0$ we conclude that

[TABLE]

Since $P=\partial_{t}Y/\partial_{t}X$, (2.5) holds. ∎

Let $T_{\varepsilon}$ be the complex space with local ring $\mathbb{C}\{\varepsilon\}/(\varepsilon^{2})$. Let $I,J$ be ideals of the ring $\mathbb{C}\{s_{1},...,s_{m}\}$. Assume $J\subset I$. Let $X,S,T$ be the germs of complex spaces with local rings $\mathbb{C}\{x,y,p\}$, $\mathbb{C}\{s\}/I,\mathbb{C}\{s\}/J$. Consider the maps $i:X\hookrightarrow X\times S$, $j:X\times S\hookrightarrow X\times T$ and $q:X\times S\to S$.

Let $\mathfrak{m}_{X},\mathfrak{m}_{S}$ be the maximal ideals of $\mathbb{C}\{x,y,p\}$, $\mathbb{C}\{s\}/I$. Let $\mathfrak{n}_{S}$ be the ideal of $\mathcal{O}_{X\times S}$ generated by $\mathfrak{m}_{X}\mathfrak{m}_{S}$.

Let $\chi:X\times S\to X\times S$ be a relative contact transformation. If $\chi$ verifies

[TABLE]

there are $\alpha,\beta,\gamma\in\mathfrak{n}_{S}$ such that

[TABLE]

Theorem 2.17**.**

$(1)$* Let $\chi:X\times S\to X\times S$ be a relative contact transformation that verifies (2.6). Then $\gamma$ is determined by $\alpha$ and $\beta$. Moreover, there is $\beta_{0}\in\mathfrak{n}_{S}+p\mathcal{O}_{X\times S}$ such that $\beta$ is the solution of the Cauchy problem*

[TABLE]

$\beta+p\mathcal{O}_{X\times S}=\beta_{0}$*.

$(2)$ Given $\alpha\in\mathfrak{n}_{S}$, $\beta_{0}\in\mathfrak{n}_{S}+p\mathcal{O}_{X\times S}$, there is a unique relative contact transformation $\chi$ that verifies (2.6) and the conditions of statement $(a)$. We denote $\chi$ by $\chi_{\alpha,\beta_{0}}$.

$(3)$ If $S=T_{\varepsilon}$ the Cauchy problem (2.8) simplifies into*

[TABLE]

Consider the contact transformations from $\mathbb{C}^{3}$ to $\mathbb{C}^{3}$ given by

[TABLE]

[TABLE]

Theorem 2.18**.**

(See [AO] or [10].) Let $\Phi:(\mathbb{C}^{3},0)\to(\mathbb{C}^{3},0)$ the the germ of a contact transformation. Then $\Phi=\Phi_{1}\Phi_{2}\Phi_{3}$, where $\Phi_{1}$ is of type $(\ref{SEMIS-3})$, $\Phi_{2}$ is of type $(\ref{PARABOLOIDAL3})$ and $\Phi_{3}$ is of type $(\ref{abc3})$, with $\alpha,\beta,\gamma\in\mathbb{C}\{x,y,p\}$. Moreover, there is $\beta_{0}\in\mathbb{C}\{x,y\}$ such that $\beta$ verifies the Cauchy problem $(\ref{E:CAUCHY-3})$, $\beta-\beta_{0}\in(p)$ and

[TABLE]

If $D\Phi(0)(\{y=p=0\})=\{y=p=0\}$, $\Phi_{2}=id_{\mathbf{C}^{3}}$.

Proposition 2.19**.**

Let $f$ and $g$ be two microlocally equivalent SQH or NND generic plane curves. Then, $f$ and $g$ have equisingular semiuniversal microlocal deformations with isomorphic base spaces.

Proof.

Let $X,Y$ denote the germs of analytic subsets at the origin of $\mathbb{C}^{3}$ defined by $Con\,f$ and $Con\,g$ respectively. Let $\chi:\mathbb{C}^{3}\to\mathbb{C}^{3}$ be a contact transformation such that $\chi(Y)=X$ and $\mathcal{X}:=(i,\Phi):X\hookrightarrow\mathbb{C}^{3}\times\mathbb{C}^{\ell}\to\mathbb{C}^{\ell}$ be a semiuniversal equisingular deformation of $X$ (to see that such an object exists see Theorem 2.15). Let us show that $(i\circ\chi,\Phi)$ is a semiuniversal equisingular deformation of $Y$:

Let $\mathcal{Y}:=(j,\Psi):Y\hookrightarrow\mathbb{C}^{3}\times\mathbb{C}^{k}\to\mathbb{C}^{k}$ be an equisingular deformation of $Y$. Because $\mathcal{X}$ is versal there is $\varphi:\mathbb{C}^{k}\to\mathbb{C}^{\ell}$ such that $\varphi^{\ast}\mathcal{X}\cong(j\circ\chi^{-1},\Psi$).

[TABLE]

Then, $(\varphi^{\ast}i\circ\chi,\varphi^{\ast}\Phi)\cong(j,\Psi)$ which means that $\varphi^{\ast}(i\circ\chi,\Phi)\cong\mathcal{Y}$. The result follows from the fact that a semiuniversal deformation is unique up to isomorphism (see Lemma $II.1.12$ of [4]). ∎

Recall that, for a SQH or NND generic plane curve $f$, there is a semiuniversal microlocal equisingular deformation with base space $\mathbb{C}^{k}$, where $k$ is the the dimension as vector space over $\mathbb{C}$ of $I_{f}/I^{\mu}_{f}$. So, because of Proposition 2.19 and Proposition $II.2.17$ of [4], the following defines an invariant between microlocally equivalent fibers of $F$.

Definition 2.20**.**

Let $f$ be a SQH or NND generic plane curve. Then

[TABLE]

is the microlocal Tjurina number of $f$.

3. The microlocal Kodaira-Spencer map

Assume $k,n$ are coprime integers, $0<2k<n$. Set $f=y^{k}-x^{n}$, $\mu=(n-2)(k-2)$. Consider in $\mathbb{C}[x,y]$ the grading given by $o(x^{i}y^{j})=ki+nj$, $(i,j)\in\mathbb{N}^{2}$. Set $\omega=o(x^{n-2}y^{k-2})-kn$, $\varpi=o(x^{n-k}y^{k-2})-kn$, $e(x^{i}y^{j})=(i,j)\in\mathbb{N}^{2}$,

[TABLE]

Let $m_{1},\ldots,m_{\mu}$ be the family $x^{i}y^{j}$, $(i,j)\in B$, ordered by degree. Set $b=\#B_{0}$. If $\mu-b+1\leq\ell\leq\mu$, set $o(\ell)=o(m_{\ell})-kn$ and $o(s_{o(\ell)})=-o(\ell)$.

Let $A\subseteq B$. Set $I_{A}=\{\ell:e(m_{\ell})\in A_{0}\}$, $s_{A}=\left(s_{o(\ell)}\right)_{\ell\in I_{A}}$. Set $\mathbb{C}^{A}=\mathbb{C}^{\#A_{0}}$ with coordinates $s_{A}$. Notice that $I_{B}=\{\mu-b+1,\ldots,\mu\}$. Moreover,

[TABLE]

is homogeneous of degree $kn$.

Let $Y$ be the plane curve defined by $f$. Let $\Gamma$ be the conormal of $Y$. Let $\mathcal{F}_{A}$ be the deformation defined by $F_{A}$. Notice that

• •

$\mathcal{F}_{B}$ is a semiuniversal equisingular deformation of $Y$,

• •

$\mathcal{F}_{C}$ is a semiuniversal equisingular microlocal deformation of $Y$,

• •

if $C\subseteq A\subseteq B$, $\mathcal{F}_{A}$ is a complete equisingular microlocal deformation of $Y$.

Let $\Delta_{F_{A}}$ be the ideal of $\mathbb{C}[s_{A}]$ generated by $\partial_{x}F_{A}$ and $\partial_{y}F_{A}$. Assume $o(p)=n-k$ in order to guarantee that the contact form $dy-pdx$ is homogeneous.

Lemma 3.1**.**

Assume $C\subseteq A\subseteq B$ and $\gamma\geq 1$. There is $H^{\gamma}_{A}\in\mathbb{C}[s_{A}]\{x,y\}$ such that $H^{\gamma}_{A}\equiv p^{\gamma}\partial_{x}F_{A}\;mod\;I_{Con(\mathcal{F}_{A})}+\Delta_{F_{A}}$ where $H^{\gamma}_{A}$ is homogeneous of degree $\gamma(n-k)+kn-k$. If $\gamma\geq k-1$, $H^{\gamma}_{A}\in\Delta_{F_{A}}$. If $C\subseteq A^{\prime}\subseteq A\subseteq B$, $H^{\gamma}_{A^{\prime}}=H^{\gamma}_{A}|_{\mathbb{C}^{A^{\prime}}}$.

Proof.

Set $\psi_{0}=\theta$, where $\theta^{k}=-1$. There are $\psi_{i}\in(s_{A})\mathbb{C}[s_{A}]$, $i\geq 1$, such that

[TABLE]

defines a parametrization $\Phi$ of $\mathcal{F}_{A}$. Setting $P(t,s_{A})=\sum_{i\geq 0}\frac{n+i}{k}\psi_{i}t^{n-k+i}$, $X,Y,P$ defines a parametrization $\Psi$ of $Con(\mathcal{F}_{A})$. Since $x$ is homogeneous of degree $k$ and $x=t^{k}$, we assume $t$ homogeneous of degree $1$. Let us show that $Y$ is homogeneous of degree $n$. The $\mathbb{C}^{\ast}$-action acts on $\Phi$ by

[TABLE]

Since $F_{A}$ is homogeneous, for each $s_{A}$,

[TABLE]

is another parametrization of the curve defined by $(x,y)\mapsto F_{A}(x,y,s_{A})$. Since the first term of both parametrizations coincide, $\Phi_{a}=\Phi$, $a\cdot\psi_{i}=a^{-i}\psi_{i}$ and $\Phi$ is homogeneous. Therefore, $\Psi$ is homogeneous.

There is an integer $c$ such that $\Phi^{\ast}(\Delta_{F_{A}})\supset t^{c}\mathbb{C}[s_{A}]\{t\}$. Remark that $p^{\gamma}\partial_{x}F_{A}$ is homogeneous of degree $\gamma(n-k)+kn-k$. We construct $H^{\gamma}_{A}$ in the following manner. There is a monomial $ax^{i}y^{j}$, $a\in\mathbb{C}[s_{A}]$ such that the monomials of lowest $t$-order $\Phi^{\ast}(ax^{i}y^{j})$ and $\Psi^{\ast}(p^{\gamma}\partial_{x}F_{A})$ coincide. Replace $p^{\gamma}\partial_{x}F_{A}$ by $p^{\gamma}\partial_{x}F_{A}-ax^{i}y^{j}$ and iterate the procedure. After a finite number of steps we construct $H^{\gamma}_{A}$ such that

[TABLE]

Therefore,

[TABLE]

Remark that the monomial $ax^{i}y^{j}$ is homogeneous of degree $\gamma(n-k)+kn-k$. ∎

Set $\Theta_{B}=Der_{\mathbb{C}}\mathbb{C}[s_{B}]$, $\partial_{o(\ell)}=\partial_{s_{o(\ell)}}$ and $o(\partial_{o(\ell)})=o(\ell)$ for each $\ell\in I_{B}$. Assume $C\subseteq A^{\prime}\subseteq A\subseteq B$. Let $\Theta_{A,A^{\prime}}$ be the $\mathbb{C}[s_{A}]$-submodule of $\Theta_{B}$ generated by $\partial_{o(\ell)}$, $\ell\in I_{A^{\prime}}$. Set $\Theta_{A}=\Theta_{A,A}$. There are maps

[TABLE]

where $r_{A,A^{\prime}}$ is the restriction to $\mathbb{C}^{A^{\prime}}$.

Definition 3.2**.**

Let $I^{\mu}_{F}$ be the ideal of $\mathbb{C}[s_{B}][[x,y]]$ generated by $F_{B}$, $\Delta F_{B}$ and $H^{\gamma}_{B}$, $\gamma=1,\ldots,k-2$. We say that the map

[TABLE]

given by $\rho(\delta)=\delta F_{B}+I^{\mu}_{F}$ is the microlocal Kodaira-Spencer map of $f$. We will denote the kernel of $\rho$ by $\mathcal{L}_{B}$.

Assume we have defined $\mathcal{L}_{A}$. We set

[TABLE]

Let $L$ be a Lie subalgebra of $\Theta_{A}$. Consider in $\mathbb{C}^{A}$ the binary relation $\sim$ given by $p\sim q$ if there is a vector field $\delta$ of $L$ and an integral curve $\gamma$ of $\delta$ such that $p$ and $q$ are in the trajectory of $\gamma$. We denote by $L$ the equivalence relation generated by $\sim$. We say that a subset $M$ of $\mathbb{C}^{A}$ is an integral manifold of $L$ if $M$ is an equivalence class of $L$.

Assume $C\subseteq A\subseteq B$. The family $m_{\ell}$, $1\leq\ell\leq\mu$, defines a basis of the $\mathbb{C}[s_{A}]$-module

[TABLE]

Set $H^{0}_{A}=F_{A}$. The relations

[TABLE]

define $c_{\ell,\upsilon}^{\gamma}\in\mathbb{C}[s_{A}]$ for each $0\leq\gamma\leq k-2$, $1\leq\ell,\upsilon\leq\mu$. Assume $A=B$ and set

[TABLE]

If $m_{\ell}=x^{i}y^{j}$ we will also denote $\delta_{\ell}^{\gamma}$ by $\delta_{i,j}^{\gamma}$. For $1\leq\gamma\leq k-2$, set

[TABLE]

Lemma 3.3**.**

With the previous notations, we have that:

1. (1)

The vector fields $\delta^{\gamma}_{\ell}\;(\delta_{i,j}^{\gamma})$ are homogeneous of degree $\alpha^{\gamma}_{\ell}\;(\alpha^{\gamma}_{i,j})$, $0\leq\gamma\leq k-2$, $1\leq\ell\leq\mu\;((i,j)\in B)$. 2. (2)

$\delta_{i,j}^{\gamma}(0)\neq 0$* if and only if $\gamma\geq 1$, $i\leq\gamma-1$, $\gamma+j\leq k-2$.* 3. (3)

$\delta_{i,j}^{\gamma}=0$* if $\alpha^{\gamma}_{i,j}>\omega$.* 4. (4)

The Lie algebra $\mathcal{L}_{B}$ is generated as $\mathbb{C}[s_{B}]$-module by $\{\delta^{\gamma}_{\ell}:0\leq\gamma\leq k-2,\;\alpha^{\gamma}_{\ell}\leq\omega\}$. 5. (5)

If $\sigma>\varpi$, $\partial_{s_{\sigma}}\in\mathcal{L}_{B}$. 6. (6)

If $(u,v)\in B\setminus C$ there is $\delta\in\mathcal{L}_{B}$ such that $\delta=\partial_{s_{\sigma}}+\varepsilon$ is homogeneous of degree $\sigma=ku+nv-kn$, where $\varepsilon$ is a linear combination of $\partial_{s_{o(i)}}$, $i\in I_{B}$, $i>\sigma$, with coefficients in $\mathbb{C}[s_{B}]$.

Proof.

(3): Just notice that if $\alpha^{0}_{i,j}>\omega=o(m_{\mu})-kn$ then $o(m_{i,j}F_{B})>o(m_{\mu})$. Now, because $n>2k$, $o(H^{\gamma}_{B})>kn=o(F_{B})$ for any $\gamma=1,\ldots,k-2$, the result holds for $\gamma>0$.

(4): For $\gamma=0$ ($1\leq\gamma\leq k-2$) and each $\ell=1,\ldots,\mu$ such that $o(m_{\ell})\leq\omega$, we have that $\rho(\delta_{\ell}^{\gamma})=\delta_{\ell}^{\gamma}F_{B}+I^{\mu}_{F}=m_{\ell}F_{B}+I^{\mu}_{F}\;(m_{\ell}H_{B}^{\gamma}+I^{\mu}_{F})=0+I^{\mu}_{F}$. So, $\{\delta^{\gamma}_{\ell}:0\leq\gamma\leq k-2,\;\alpha^{\gamma}_{\ell}\leq\omega\}\subset\mathcal{L}_{B}$.

Now, let

[TABLE]

such that $\rho(\delta)=0$. Then

[TABLE]

with $M_{0},\ldots,M_{k-2}\in\mathbb{C}[s_{B}][[x,y]]$. Suppose

[TABLE]

where the $M_{\gamma,\ell}\in\mathbb{C}[s_{B}]$ for each $\ell=1,\ldots,\mu,\,\gamma=0,\ldots,k-2$. Then

[TABLE]

Similarly, for any $\gamma=1,\ldots,k-2$

[TABLE]

So,

[TABLE]

which means that

[TABLE]

∎

Let $L_{B}$ be the Lie algebra generated by $\delta^{\gamma}_{\ell}$, $\gamma=0,\ldots,k-2$, $\ell=1,\ldots,b$. Remark that $\mathbb{C}^{B}/\mathcal{L}_{B}\cong\mathbb{C}^{B}/L_{B}$. Consider a matrix with lines given by the coefficients of the vector fields $\delta^{\gamma}_{\ell}$, $\gamma=0,\ldots,k-2$, $\ell=1,\ldots,b$. After performing Gaussian diagonalization we can assume that:

• •

For each $\sigma\in I_{B}\setminus I_{C}$ there is a line corresponding to a vector field $\partial_{s_{o(\sigma)}}+\varepsilon$, where $\varepsilon\in\Theta_{B,C}$.

• •

The remaining lines correspond to vector fields $\delta^{\prime}_{\ell}$, $\ell\in J$, of $\Theta_{B,C}$.

The vector fields $\delta^{\prime}_{\ell}$, $\ell\in J$, generate $\mathcal{L}_{B,C}$ as a $\mathbb{C}[s_{B}]$-module. Let $\delta_{\ell}$ be the restriction of $\delta^{\prime}_{\ell}$ to $\mathbb{C}^{C}$ for each $\ell\in J$. The vector fields $\delta_{\ell},\;\ell\in J$, generate $\mathcal{L}_{C}$ as $\mathbb{C}[s_{C}]$-module. Note that $\{\delta_{\ell},\;\ell\in J\}$ is in general not uniquely determined but the $\mathbb{C}[s_{C}]$-module generated by them is. Let $L_{C}$ be the Lie algebra generated by $\{\delta_{\ell},\;\ell\in J\}$. Since $L_{C}\subseteq L_{B}$ the inclusion map $\mathbb{C}^{C}\hookrightarrow\mathbb{C}^{B}$ defines a map $\mathbb{C}^{C}/L_{C}\to\mathbb{C}^{B}/L_{B}$. By statement $(6)$ of Lemma 3.3, this map is surjective.

Assume there is a vector field $\delta_{\ell}$, $\ell\in J$, of order $\alpha$. Let $\{\delta^{\alpha,i}:i\in I_{\alpha}\}$ be the set of vector fields $\delta_{\ell}$, $\ell\in J$, of order $\alpha$, with $I_{\alpha}=\{1,\ldots,\#I_{\alpha}\}$. If there is $\ell_{0}$ such that $\delta^{\alpha,j}(s_{\ell})=0$ for $\ell\leq\ell_{0}$ and $\delta^{\alpha,i}(s_{\ell_{0}})\neq 0$, we assume that $i<j$. If $I_{\alpha}=\{1\}$, set $\delta^{\alpha}=\delta^{\alpha,1}$.

Remark 3.4*.*

If $k=7,n=15$, we have that a semiuniversal equisingular microlocal deformation of $f$ given by

[TABLE]

Notice that the vector fields $\delta^{0}_{0,1}$ and $\delta^{1}_{2,0}$ give origin to the linearly independent vector fields

[TABLE]

and

[TABLE]

Theorem 3.5**.**

The map $\mathbb{C}^{C}/L_{C}\to\mathbb{C}^{B}/L_{B}$ is bijective.

Proof.

Let $I_{p}$ be the subset of $I_{B}$ that contains $I_{C}$ and the $p$ smallest elements of $I_{B}\setminus I_{C}$. Set $C_{p}=\{(i,j)\in B:ki+nj-kn\in I_{p}\}$. The Lie algebra $L_{C_{p}}=\mathcal{L}_{C_{p}}\cup L_{B}$ generates $\mathcal{L}_{C_{p}}$ as $\mathbb{C}[s_{C_{p}}]$-module. There is $p$ such that $C_{p}=D$. By statement $(5)$ of Lemma 3.3 the integral manifolds of $L_{B}$ are of the type $M\times\mathbb{C}^{B\setminus D}$, where $M$ is an integral manifold of $L_{C_{p}}$. Therefore, $\mathbb{C}^{D}/L_{D}\cong\mathbb{C}^{B}/L_{B}$. Assume $\mathbb{C}^{C_{p+1}}/L_{C_{p+1}}\cong\mathbb{C}^{B}/L_{B}$ and $I_{C_{p+1}}\setminus{C_{p}}=\{\sigma\}$. The Lie algebra $\mathcal{L}_{C_{p+1}}$ is generated by $\mathcal{L}_{C_{p}}$ and a vector field $\partial_{s_{o(\sigma)}}+\varepsilon$, where $\varepsilon\in\mathcal{L}_{C_{p+1},C}$. Consider the flow of $\partial_{s_{o(\sigma)}}+\varepsilon$ with initial condition at a point of $\mathbb{C}^{C_{p}}$. We can use this flow to construct an homogeneous affine isomorphism of $\mathbb{C}^{C_{p+1}}$ into itself that equals the identity on $\mathbb{C}^{C_{p}}$ and rectifies $\partial_{s_{o(\sigma)}}+\varepsilon$, leaving invariant $\mathcal{L}_{C_{p}}$. Hence, $\mathbb{C}^{C_{p}}/L_{C_{p}}\cong\mathbb{C}^{C_{p+1}}/L_{C_{p+1}}$.

∎

Remark 3.6*.*

Let us denote by $P(s_{C})$ the restriction of $P\in\mathbb{C}[s_{B}][[x,y]]$ to $\mathbb{C}^{C}$. Then, $F_{B}(s_{C})=F_{C}$, $\Delta F_{B}(s_{C})=\Delta F_{C}$ and $H^{\gamma}_{B}(s_{C})=H^{\gamma}_{C}$ for each $\gamma=1,\ldots,k-2$. Let $\{\delta_{\ell,\mu},\;\ell\in J\}\subset Der_{\mathbb{C}}\,\mathbb{C}[s_{C}]$ be the set of vector fields obtained if we proceed as in the definition of $\{\delta^{\prime}_{\ell},\;\ell\in J\}$, now with $C$ in the place of $B$. Then $<\{\delta_{\ell,\mu}\}>$=$<\{\delta_{\ell}\}>$ as $\mathbb{C}[s_{C}]$-modules. To see this just notice that, if

[TABLE]

then

[TABLE]

4. Geometric Quotients of Unipotent Group Actions

An affine algebraic group is said to be unipotent if it is isomorphic to a group of upper triangular matrices of the form $Id+\varepsilon$, where $\varepsilon$ is nilpotent. If $\mathcal{G}$ is unipotent its Lie algebra $L$ is nilpotent and the map $exp:L\to\mathcal{G}$ is algebraic. Given a nilpotent Lie algebra $L$, there is a unipotent group $\mathcal{G}=exp\;L$ such that $L$ is the Lie algebra of $\mathcal{G}$.

Let $A$ be a Noetherian $\mathbb{C}$-algebra. A linear map $D:A\to A$ is a derivation of $A$ if $D(fg)=fD(g)+gD(f)$. A derivation $D$ of $A$ is nilpotent if for each $f\in A$ there is $n$ such that $D^{n}(f)=0$. Let $Der^{nil}(A)$ denote the Lie algebra of nilpotent derivations of $A$. Here, we set $A=\mathbb{C}[s_{C}]$.

Let $\mathcal{G}$ be an algebraic group acting algebraically on an algebraic variety $X$. If $Y$ is an algebraic variety and $\pi:X\to Y$ a morphism then $\pi$ is called a geometric quotient, if

1. (1)

$\pi$ is surjective and open, 2. (2)

$(\pi_{\ast}\mathcal{O}_{X})^{\mathcal{G}}=\mathcal{O}_{Y}$, 3. (3)

$\pi$ is a orbit map, i.e. the fibres of $\pi$ are orbits of $\mathcal{G}$.

If a geometric quotient exists it is uniquely determined and we just say that $X/\mathcal{G}$ exists. Here, $\mathcal{G}$ will act on each strata of $\mathbb{C}^{c}=Spec\,A$ through the action of $\mathcal{G}$ on each fiber of $G$. On Theorem 5.3 we prove that $\mathbb{C}^{c}/\L_{c}$ is a classifying space for germs of Legendrian curves with generic plane projection $\{y^{k}+x^{n}=0\}$. The integral manifolds of ${L_{C}}$ are the orbits of the action of $\mathcal{G}_{0}:=exp\,{L_{C}}$. Set $L:=[{L_{C}},{L_{C}}]$ and $\mathcal{G}=exp\,L$. Note that $L$ is nilpotent ($\mathcal{G}$ unipotent) and ${L_{C}}/L\cong\mathbb{C}\delta_{0}$, where $\delta_{0}$ is the Euler field.

Definition 4.1**.**

Let $\mathcal{G}$ be a unipotent algebraic group, $Z=Spec\,A$ an affine $\mathcal{G}$-variety and $X\subseteq Z$ open and $\mathcal{G}$-stable. Let $\pi:X\to Y:=Spec\,A^{\mathcal{G}}$ be the canonical map. A point $x\in X$ is called stable under the action of $\mathcal{G}$ with respect to $A$ (or with respect to $Z$) if the following holds:

There exists an $f\in A^{\mathcal{G}}$ such that $x\in X_{f}=\{y\in X,f(y)\neq 0\}$ and $\pi:X_{f}\to Y_{f}:=Spec\,A_{f}^{\mathcal{G}}$ is open and an orbit map. If $X=Z=Spec\,A$ we call a point stable with respect to $A$ just stable.

Let $X^{s}(A)$ denote the set of stable points of $X$ (under $\mathcal{G}$ with respect to $A$).

Proposition 4.2** ([6]).**

With the previous notations, we have that:

1. (1)

$X^{s}(A)$* is open and $\mathcal{G}$-stable.* 2. (2)

$X^{s}(A)/\mathcal{G}$* exists and is a quasiaffine algebraic variety.* 3. (3)

If $V\subset Spec\,A^{\mathcal{G}}$ is open, $U=\pi^{-1}(V)$ and $\pi:U\to V$ is a geometric quotient then $U\subset X^{s}(A)$. 4. (4)

If $X$ is reduced then $X^{s}(A)$ is dense in $X$.

Definition 4.3**.**

A geometric quotient $\pi:X\to Y$ is locally trivial if an open covering $\{V_{i}\}_{i\in I}$ of $Y$ and $n_{i}\geq 0$ exist, such that $\pi^{-1}(V_{i})\cong V_{i}\times A_{\mathbb{C}}^{n_{i}}$ over $V_{i}$.

We use the following notations:

Let $L\subseteq Der^{nil}(A)$ be a nilpotent Lie-algebra and $d:A\to Hom_{\mathbb{C}}(L,A)$ the differential defined by $da(\delta)=\delta(a)$. If $B\subset A$ is a subalgebra then $\int B:=\{a\in A:\delta(a)\in B\;\text{for all}\;\delta\in L\}$. If $\mathfrak{a}\subset A$ is an ideal, $V(\mathfrak{a})$ denotes the closed subscheme $Spec\,A/\mathfrak{a}$ of $Spec\,A$ and $D(\mathfrak{a})$ the open subscheme $Spec\,A-V(\mathfrak{a})$.

Let $A$ be a noetherian $\mathbb{C}$-algebra and $L\subseteq Der^{nil}(A)$ a finite dimensional nilpotent Lie-algebra. Suppose that $A=\cup_{i\in\mathbb{Z}}F^{i}(A)$ has a filtration

[TABLE]

by sub-vector spaces $F^{i}(A)$ such that

[TABLE]

Assume, furthermore, that

[TABLE]

is filtered by sub-Lie-algebras $Z_{j}(L)$ such that

[TABLE]

The filtration $Z_{\bullet}$ of $L$ induces projections

[TABLE]

For a point $t\in Spec\,A$ with residue field $\kappa(t)$ let

[TABLE]

with $\rho$ minimal such that $AdF^{\rho}(A)=AdA$,

[TABLE]

such that $s_{j}(t)$ is the orbit dimension of $Z_{j}(L)$ at $t$.

Let $Spec\,A=\cup U_{\alpha}$ be the flattening stratification of the modules

[TABLE]

and

[TABLE]

Theorem 4.4** ([6]).**

Each stratum $U_{\alpha}$ is invariant by $L$ and admits a locally trivial geometric quotient with respect to the action of $L$. The functions $r_{i}(t)$ and $s_{i}(t)$ are constant along $U_{\alpha}$. Let $x_{1},\ldots,x_{p}\in A,\delta_{1},\ldots,\delta_{q}\in L$ satisfying the following properties:

• •

there are $\nu_{1},\ldots,\nu_{\rho},0\leq\nu_{1}<\ldots<\nu_{\rho}=p$, such that $dx_{1},\ldots,dx_{\nu_{i}}$ generate the $A$-module $AdF^{i}(A)$;

• •

there are $\mu_{0},\ldots,\mu_{\ell},1=\mu_{0}<\mu_{1}<\ldots<\mu_{\ell}$ such that $\delta_{\mu_{j}},\ldots,\delta_{m}\in Z_{j}(L)$ and $Z_{j}(L)\subseteq\sum_{i\geq\mu_{j}}A\delta_{i}$.

Then

[TABLE]

The strata $U_{\alpha}$ are defined set theoretically by fixing (4.1) and (4.2).

5. Filtrations and Strata

Set $L=[L_{C},L_{C}]$. Fix a integer $a$ such that $k\geq a\geq 0$. For each $i\in\mathbb{Z}$ let $F^{i}_{a}$ be the $\mathbb{C}$-vector space generated by monomials in $\mathbb{C}[s_{C}]$ of degree $\geq-(a+ik)$. Since $o(\delta)\geq k$ for each homogeneous vector field of $L$, $LF^{j}_{a}\subseteq F^{j-1}_{a}$ for each $j$. For each $m\in\mathbb{Z}$ let $I^{m}_{a}$ be the ideal of $\mathbb{C}[[x,y]]$ generated by the monomials of degree $\geq a+mk$. Let $\rho$ be the smallest $i$ such that $dF^{i}_{a}$ generates $\mathbb{C}[s_{C}]d\mathbb{C}[s_{C}]$ as a $\mathbb{C}[s_{C}]$-module.

Given $\alpha\in\mathbb{Z}$, set $\alpha^{\vee}:=nk-k^{2}-2n-\alpha$. For each integer $j$ set $S_{j}=\{\alpha:s_{\alpha^{\vee}}\in F^{\rho-j}_{a},\,\alpha\neq 0\}$ and let $Z^{a}_{j}$ be the sub-Lie algebra of $L$ generated by the homogeneous vector fields $\delta\in L$ such that $o(\delta)\in S_{j}$. Remark that

[TABLE]

For each $t\in\mathbb{C}^{C}$ let $I^{\mu}_{t}$ be the ideal of $\mathbb{C}[[x,y]]$ generated by $F_{t},\Delta F_{t}$ and $H^{1}_{t},\ldots,H^{k-2}_{t}$. Set

[TABLE]

for $m=n,\ldots,n+\rho$ and

[TABLE]

We say that $\widehat{\tau}_{a}^{\bullet}(t)$ is the microlocal Hilbert function of $X_{t}$. Set

[TABLE]

We only define $\widehat{\tau}_{a}^{\bullet}(t)$ for $m=n,\ldots,n+\rho$ because

[TABLE]

(the microlocal Tjurina number of $X_{t}$) if $m$ is big and

[TABLE]

(hence independent of $t$) if $m$ is small.

Let $\{U^{a}_{\alpha}\}$ be the flattening stratification of $\mathbb{C}^{C}$ corresponding to $F^{\bullet}_{a}$ and $Z^{a}_{\bullet}$. It follows from Theorem 4.4 that $U^{a}_{\alpha}\to U^{a}_{\alpha}/L$ is a geometric quotient. Moreover, ${L_{C}}/L\cong\mathbb{C}^{\ast}$ acts on $U^{a}_{\alpha}/L$ and $U^{a}_{\alpha}/\mathcal{L}_{C}=U^{a}_{\alpha}/{L_{C}}$ is a geometric quotient of $U^{a}_{\alpha}$ by ${L_{C}}$. For $t\in\mathbb{C}^{C}$ let us define

[TABLE]

where

[TABLE]

and

[TABLE]

Lemma 5.1**.**

The function $t\mapsto\underline{e}^{a}(t)$ is constant on $U^{a}_{\alpha}$ and takes different values for different $\alpha$. The analytic structure of $U^{a}_{\alpha}$ is defined by the corresponding subminors of $(\delta({s_{C}})(t))$. Moreover, $u^{a}_{j}(t)=\widehat{\mu}_{1}^{n+j}-\widehat{\tau}_{a,1}^{n+j}(t)$ and $v^{a}_{j}(t)=\widehat{\mu}_{2}^{n+j}-\widehat{\tau}_{a,2}^{n+j}(t)$. In particular, $u^{a}_{\rho}(t)=v^{a}_{\rho}(t)=\widehat{\mu}-\widehat{\tau}(X_{t})$ where $\widehat{\tau}(X_{t})$ is the microlocal Tjurina number of the curve singularity $X_{t}$.

Proof.

That $\underline{e}^{a}(t)$ is constant on $U^{a}_{\alpha}$ and takes different values for different $\alpha$ is a consequence of Theorem, 4.4, as is the claim about the analytic structure of each strata.

Let $t\in U^{a}_{\alpha}$ and consider for each $m\in\{n,\ldots,n+\rho\}$ the induced $\mathbb{C}$-base $\{m_{\ell\in J_{m}}(t)\}=\{m_{\ell\in J_{m}}\}$ of $\mathbb{C}\{x,y\}/(\Delta F_{t},I^{m}_{a})$. Then, for each $\ell\in J_{m}$

[TABLE]

and

[TABLE]

for $\gamma=1,\ldots,k-2$. Then, by definition of $\widehat{\tau}_{a}^{\bullet}(t)$ and from the definition of $\{\delta_{\mu}\}$, $u^{a}_{j}(t)=\widehat{\mu}_{1}^{n+j}-\widehat{\tau}_{a,1}^{n+j}(t)$.

The proof of the claim about the $v^{a}_{j}(t)$ is similar with the difference that we’re now interested in the relations $mod\;\Delta F_{t}$ between the $m_{\ell}F_{t},m_{\ell}H_{t}^{\gamma}$ that belong to $I^{\rho-1+2n-m}_{a}$ for each $m\in\{n,\ldots,n+\rho\}$. Note that $m_{\ell}F_{t},m_{\ell}H_{t}^{\gamma}\in I^{\rho-1+2n-m}_{a}$ if and only if $\alpha^{0}_{\ell},\alpha^{\gamma}_{\ell}\in S_{m-n}$.

∎

Lemma 5.2**.**

If $a,b\in\mathbb{C}^{B}$ are such that $\mathcal{C}on(\mathcal{F}_{a})\cong\mathcal{C}on(\mathcal{F}_{b})$, there is $\psi:\mathbb{C}\to\mathbb{C}^{B}$ microlocally trivial such that $\psi(0)=a$ and $\psi(1)=b$.

Proof.

Let $\chi_{0}$ be a contact transformation given by $\alpha,\beta_{0}$ such that $F_{b}=uF^{\chi_{0}}_{a}$ for some unit $u\in\mathbb{C}\{x,y\}$. We can assume $deg\;\chi_{0}>0$. There is a relative contact transformation $\chi(t)$ over $\mathbb{C}$ such that $\chi(0)=id_{\mathbb{C}^{3}}$ and $\chi(1)=\chi_{0}$. Then

[TABLE]

is an unfolding of $F_{a}$ such that $G(1)=F_{b}$. By versality of $F_{B}$ and because $F_{a}$ is semiquasihomogeneous ($j(F_{a})=(F_{a},j(F_{a}))$) there is a relative coordinate transformation

[TABLE]

and $\psi:\mathbb{C}\to\mathbb{C}^{B}$ such that

[TABLE]

(see Remark $1.1$ and Corollary $3.3$ of [MSSS]). Now, because $F_{B}$ is semiuniversal (hence does not contain trivial subfamilies with respect to right equivalence) $\Phi(1)\left(G(1)\right)=\Phi(1)\left(F_{b}\right)=F_{\psi(1)}$ implies that $\psi(1)=b$.

∎

Theorem 5.3**.**

Given $a,b\in\mathbb{C}^{C}$, $\mathcal{C}on(\mathcal{F}_{a})\cong\mathcal{C}on(\mathcal{F}_{b})$ if and only if $a$ and $b$ are in the same integral manifold of $\mathcal{L}_{C}$.

Proof.

By Theorem 3.5 we can replace $C$ by $B$.

Let us first prove sufficiency. Let $C\subset A\subset B$ and $S$ be a complex space. We say that a holomorphic map $\psi:S\to\mathbb{C}^{A}$ is trivial if for each $o\in S$, $\psi^{\ast}\mathcal{F}_{A}$ is a trivial deformation of $\mathcal{D}ef^{\,es,\mu}_{f}(S,o)$. Assume $\psi:(\mathbb{C},0)\to\mathbb{C}^{B}$ is the germ of an integral curve of a vector field $\delta$ in $\mathcal{L}_{B}$. Set $q=\psi(0)$. Let $\psi_{\varepsilon}:T_{\varepsilon}\to\mathbb{C}^{B}$ be the morphism induced by $\psi$. There are $a_{0},a_{1},\ldots,a_{l},\alpha_{0},\beta_{0}\in\mathbb{C}\{s_{B}\}[[x,y]]$ such that

[TABLE]

Set $u=1+\varepsilon a_{0}(q)$, $\alpha=\alpha(q)+\sum_{j=1}^{\ell}a_{j}(q)p^{j}$ and $\beta=\beta(q)+\sum_{j=1}^{\ell}\frac{j}{j+1}a_{j}(q)p^{j+1}$. By Theorem 2.17 there is $\gamma\in\mathbb{C}\{x,y,p\}$ such that

[TABLE]

defines a relative contact transformation $\chi^{\varepsilon}$ over $T_{\varepsilon}$. Let $G\in\mathbb{C}\{x,y,p,\varepsilon\}$ be defined by $G(x,y,p,\varepsilon)=F_{B}(x+\alpha\varepsilon,y+\beta\varepsilon,q)$. Since $\psi^{\ast}\mathcal{F}_{B}\equiv uG\;mod\;(\varepsilon)$ and

[TABLE]

we have that

[TABLE]

Therefore, $\psi_{\varepsilon}^{\ast}F_{B}$ is a trivial deformation of $\mathcal{D}ef^{\,es,\mu}_{f}(T_{\varepsilon})$. Then $\Psi^{\ast}F_{B}$ is a trivial deformation of $\mathcal{D}ef^{\,es,\mu}_{f}(\mathbb{C},0)$ (see the proof of Theorem 2.15).

Conversely, assume that there is a germ of contact transformation $\chi_{1}$ such that $(F_{a}^{\chi_{1}})=(F_{b})$. We can assume $deg\;\chi_{1}>0$. If $\chi_{1}$ is of type (2.10), by Lemma 5.2 there is a trivial curve $\psi:\mathbb{C}\to\mathbb{C}^{B}$ such that $\psi(0)=a$ and $\psi(1)=b$. Moreover, $\psi$ is an integral curve of the Euler vector field. Since the derivative of $\chi_{1}$ leaves $\{y=p=0\}$ invariant, we can assume by Theorem 2.18 that $\chi_{1}$ is of type (2.7). Set $\chi=\chi_{t\alpha,t\beta_{0}}$. There is a curve with polynomial coefficients $\psi:\mathbb{C}\to\mathbb{C}^{B}$ such that $F_{a}^{\chi}=\psi^{\ast}F_{B}$, $\psi(0)=a$ and $\psi(1)=b$.

Let $\Omega$ be an open set of $\mathbb{C}$. Let $\psi:\Omega\to\mathbb{C}^{B}$ be a trivial curve. Let us show that $\psi$ is contained in an integral manifold of $\mathcal{L}_{B}$. Let $U$ be the union of the strata $U_{\alpha}$ such that, for each $c\in U$ the microlocal Tjurina number of $F_{c}$ equals the microlocal Tjurina number of $F_{a}$. Remark that the trajectory of $\psi$ is contained in $U$. By Theorem 4.4 $\mathcal{L}_{B|U}$ verifies the Frobenius Theorem. Hence, it is enough to show that, for each $t_{0}\in\Omega$, there is $\delta\in\mathcal{L}_{B}$ such that $\psi^{\prime}(t_{0})=\delta(\psi(t_{0}))$. We can assume $t_{0}=0$. Since $\psi$ is trivial, there are a relative contact transformation $\chi$ and $u\in\mathbb{C}\{x,y,t\}$ such that $u(x,y,0)=1$ and

[TABLE]

If $\chi$ is of type 2.10, we can assume $\delta$ is the Euler field. Hence we can assume that $\chi$ is of type (2.7). Therefore there are $\ell\geq 1$ and $a,b,a_{i}\in\mathbb{C}\{x,y\}$, $1\leq i\leq\ell$, such that

[TABLE]

Deriving in order to $t$ and evaluating at [math], there is $a_{0}\in\mathbb{C}\{x,y\}$ such that

[TABLE]

There are $\delta\in L_{B}$ and $\varepsilon\in\Delta_{F_{B}}$ such that

[TABLE]

Hence

[TABLE]

If $\delta=\sum_{(i,j)\in B_{0}}a_{i,j}\partial_{s_{i,j}}$, $a_{i,j}(\psi(0))=\psi^{\prime}_{i,j}(0)$ for each $(i,j)\in B_{0}$. ∎

Theorem 5.4**.**

$(1)$* Let $\underline{e}=(e_{1},\ldots,e_{\rho})\in\mathbb{N}^{\rho+1}$ and let $U^{a}_{\underline{e}}$ denote the unique stratum (assumed to be not empty) such that $\underline{e}^{a}(t)=\underline{e}$ for each $t\in U^{a}_{\underline{e}}$. The geometric quotient $U^{a}_{\underline{e}}/\mathcal{L}$ is quasiaffine and of finite type over $\mathbb{C}$. It is a coarse moduli space for the functor which associates to any complex space $S$ the set of isomorphism classes of flat families (with section) over $S$ of plane curve singularities with fixed semigroup $\langle k,n\rangle$ and fixed microlocal Hilbert function $\widehat{\tau}_{a}^{\bullet}$.

$(2)$ Let $T_{\widehat{\tau}_{min}}$ be the open dense set defined by singularities with minimal microlocal Tjurina number $\widehat{\tau}_{min}$. Then the geometric quotient $T_{\widehat{\tau}_{min}}/\mathcal{L}_{C}$ exists and is a coarse moduli space for curves with semigroup $\langle k,n\rangle$ and microlocal Tjurina number $\widehat{\tau}_{min}$. Moreover, $T_{\widehat{\tau}_{min}}/\mathcal{L}_{C}$ is locally isomorphic to an open subset of a weighted projective space.*

Proof.

It follows from Lemma 5.1 and Theorems 4.4 and 5.3. ∎

6. Example

The function

[TABLE]

is a semiuniversal equisingular microlocal deformation of $f=y^{6}+x^{13}$.

The Lie algebra $L_{C}$ is generated by the vector fields

[TABLE]

Choosing $a=6$ we get $F^{0}_{a}=\langle s_{2},s_{3},s_{4}\rangle$, $F^{1}_{a}=\langle s_{2},s_{3},s_{4},s_{9},s_{10}\rangle$, $F^{2}_{a}=\langle s_{2},s_{3},s_{4},s_{9},s_{10},s_{16}\rangle$. So, $\rho=2$ and the stratification $\{U^{a}_{\alpha}\}$ given by fixing $\underline{e}^{a}(t)=(u^{a}_{0}(t),u^{a}_{1}(t),u^{a}_{2}(t);v^{a}_{0}(t),v^{a}_{1}(t),v^{a}_{2}(t))$ is given by

[TABLE]

$U_{1}$ is the stratum with minimal microlocal Tjurina number.

Let us present detailed calculations concerning the generators of $L_{C}$ in the previous example. Let $Y$ denote the germ at the origin of $\{F_{C}=0\}$. The relative conormal $\mathcal{L}$ of $Y$ can be parametrized by

[TABLE]

where $\psi_{i}\in({s_{C}})\mathbb{C}[{s_{C}}]$ are homogeneous of degree $-i$. These are the $a_{i}$ such that the polynomial in $\mathbb{C}[t]$ given by the following SINGULAR session is zero:

ring r=(0,a2,a3,a4,a5,a6,a7,a8,a9,s2,s3,s4,s9,s10,s16),(x,y,t),dp;

poly F=y6+x13+s2x9y2+s3x7y3+s4x5y4+s9x8y3+s10x6y4+s16x7y4;

subst(F,x,-t6);

-t7̂8 +(-s2)y2̂t5̂4+(s9)y3̂t4̂8+(-s16)y4̂t4̂2+(-s3)y3̂t4̂2

+(s10)y4̂t3̂6+(-s4)y4̂t3̂0+y6̂

subst(-t7̂8 +(-s2)y2̂t5̂4+(s9)y3̂t4̂8+(-s16)y4̂t4̂2+(-s3)y3̂t4̂2

+(s10)y4̂t3̂6+(-s4)y4̂t3̂0+y6̂,y,t1̂3+a2t1̂5+a3t1̂6+a4t1̂7+a5t1̂8

+a6t1̂9+a7t2̂0+a8t2̂1+a9t2̂2)\

As we’ll see, the only $\psi_{i}$ we actually need to find the generators of ${L_{C}}$ is

[TABLE]

Let us calculate the vector fields generating ${L_{C}}$. Here, all equalities are $mod\,\Delta F_{C}$ and in the vector fields we identify, by abuse of language, the monomials and the corresponding $\partial$’s :

• •

$\delta^{14}$:

[TABLE]

Notice that, as a consequence of Lemma 3.3, the monomials occurring with order bigger than $deg(x^{7}y^{4})$ can be ignored in this calculation. From now on, whenever we use the symbol $\cdots$ we mean that bigger order monomials can be ignored. Now, continuing the previous SINGULAR session:

poly p=(-13t7-15a2t9-16a3t10-17a4t11-18a5t12-19a6t13-20a7t14

-21a8t15-22a9t16)/6;

poly X=-t6;

poly Y=t13+a2t15+a3t16+a4t17+a5t18+a6t19+a7t20+a8t21+a9t22;

p2̂X1̂3-(13/6)2̂X1̂1Y2̂;\ (-35a92̂)/4t1̂10+(-293a8a9)/18t1̂09+(-271a7a9-136a82̂)/18t1̂08+(-249a6a9-251a7a8)/18t1̂07+(-454a5a9-460a6a8-231a72̂)/36t1̂06+(-205a4a9-209a5a8-211a6a7)/18t1̂05+(-183a3a9-188a4a8-191a5a7-96a62̂)/18t1̂04+(-161a2a9-167a3a8-171a4a7-173a5a6)/18t1̂03+(-292a2a8-302a3a7-308a4a6-155a52̂)/36t1̂02+(-131a2a7-135a3a6-137a4a5-117a9)/18t1̂01+(-116a2a6-119a3a5-60a42̂-104a8)/18t1̂00 +(-101a2a5-103a3a4-91a7)/18t9̂9+(-172a2a4-87a32̂-156a6)/36t9̂8\ +(-71a2a3-65a5)/18t9̂7+(-14a22̂-26a4)/9t9̂6+(-13a3)/6t9̂5+(-13a2)/9t9̂4

we see that

[TABLE]

Now, $\delta^{1}_{3}$, given by $yH_{1}$, which has the same order as $xH_{2}$ can be used to, through elementary operations, eliminate from $\delta^{2}_{1}$ the monomial $x^{11}y^{2}$. Thus,

[TABLE]

• •

$\delta^{13}$:

[TABLE]

But, as

[TABLE]

we see that, through elementary operations involving $\delta^{3}_{0}$, we can eliminate from $\delta^{0}_{3}$ the monomial $x^{9}y^{3}$. Thus,

[TABLE]

• •

$\delta^{12}$:

[TABLE]

through elementary operations involving $\delta^{1}_{3}$ and $\delta^{3}_{0}$ we can eliminate the monomials $x^{11}y^{2}$ and $x^{9}y^{3}$ from $\delta^{0}_{2}$ and get:

[TABLE]

Finally, using $\delta^{14}$ to eliminate $\ast s_{2}^{2}x^{7}y^{4}$, we have that

[TABLE]

• •

$\delta^{7}$:

[TABLE]

and

[TABLE]

Remark 6.1*.*

The reason why we can ignore in $px^{13}$ the monomials that occur after $x^{6}y^{4}$ is that

1. (1)

All monomials after $x^{6}y^{4}$, except for $x^{7}y^{4}$, can be eliminated because of Lemma 3.3 and through elementary operations involving $\delta^{1}_{3}$ and $\delta^{3}_{0}$. 2. (2)

Even $x^{7}y^{4}$ can be ignored, observing that $px^{13}$ is homogeneous of degree $7$ and as such, the only variables involved in the coefficient (in $\mathbb{C}[s_{C}]$) of $x^{7}y^{4}$ may be $s_{2}$, $s_{3}$ or $s_{4}$. Now, using $\delta^{14}$, $\delta^{13}$ and $\delta^{12}$ we can eliminate, through elementary operations, the monomial $x^{7}y^{4}$ from $\delta^{7}$.

From

[TABLE]

we get that

[TABLE]

Reasoning as in remark 6.1 we see that $s_{4}x^{4}y^{5}$ can be ignored. Thus,

[TABLE]

Now,

[TABLE]

[TABLE]

[TABLE]

Once again, the monomials ignored can be eliminated, reasoning as in Remark 6.1. So,

[TABLE]

We get that

[TABLE]

• •

$\delta^{6}$:

[TABLE]

Because (monomials ignored as in Remark 6.1)

[TABLE]

[TABLE]

and

[TABLE]

wet get

[TABLE]

So,

[TABLE]