Unimodular ICIS, a Classifier

TL;DR

This paper provides algorithms and a classifier implementation for identifying unimodular complete intersection surface singularities, completing the classification previously only partially addressed by Wall.

Contribution

It introduces a complete list of unimodular surface singularities and a computational classifier implemented in Singular for their identification.

Findings

Algorithms for normal form computation of unimodular singularities.

A complete classification list of unimodular surface singularities.

Implementation of a classifier in Singular software.

Abstract

We present the algorithms for computing the normal form of unimodular complete intersection surface singularities classified by C. T. C. Wall. He indicated in the list only the $\mu$-constant strata and not the complete classification in each case. We give a complete list of surface unimodular singularities. We also give the description of a classifier which is implemented in computer algebra system \singular.

Tables

Table 1. Table 1. Charecterization table of normal forms of 2 − j ​ e ​ t 2 𝑗 𝑒 𝑡 2-jet of I 𝐼 I

Table 2. Table 2.

Table 3. Table 3.

Table 4. Table 4.

Table 5. Table 5.

Table 6. Table 6.

Table 7. Table 7.

Table 8. Table 8.

Table 9. Table 9.

Table 10. Table 10.

Table 11. Table 11.

Table 12. Table 12.

Equations

$$l_i(x,y)=\{\begin{array}{ll}x{y}^q,& \text{if }i=2q\\ y^{q+2},& \text{if }i=2q+1\end{array}$$

$$⟨xy-xz+w^3,yz-xy+\lambda w^3⟩$$

$$⟨xy-xz+w^3,yz-xy+\lambda w^3+w^4⟩.$$

$$⟨xy-xz+w^3,yz-xy+\lambda w^3⟩$$

$$⟨xy-xz+w^3+t_{1}zw+t_{2}w+t_{3}z+t_4,yz-xy+\lambda w^3+{\lambda }_1{w}^4+{\lambda }_2{w}^3+{\lambda }_3{w}^2+{\lambda }_{4}zw+{\lambda }_{5}yw+{\lambda }_{6}w+{\lambda }_{7}z+{\lambda }_{8}y+{\lambda }_9⟩.$$

$$⟨xy-xz+w^3,yz-xy+\lambda w^3+{\lambda }_1{w}^4⟩.$$

$$⟨xy-xz+w^3,yz-xy+\lambda w^3+{\lambda }_1{\xi }^4{w}^4⟩.$$

$$⟨xy-xz+w^3,yz-xy+\lambda w^3+w^4⟩$$

$$⟨xy+z^2,xz+w^2+y^{3m+3}⟩$$

$$⟨xy+z^2,xz+w^2+y^{3m+3}+z{y}^{3m+2-i}⟩fori=0,1,\dots ,m.$$

$$⟨xy+z^2,xz+w^2+y^{3m+3}⟩.$$

$$⟨xy+z^2+Aw+Bz+C,w^2+xz+y^{3m+3}+\sum _{i=0}^{3m+2}{\alpha }_i{y}^{3m+2-i}z+\sum _{i=0}^{3m+2}{\beta }_i{y}^{3m+2-i}⟩.$$

$$(w_1,w_2,w_3,w_4)=(12m+10,6,6m+8,9m+9)$$

$$(d_1,d_2)=(12m+16,18m+18).$$

$$⟨xy+z^2,w^2+xz+y^{3m+3}+\sum _{i=0}^m{\alpha }_i{y}^{3m+2-i}z⟩.$$

$$I=⟨xy+z^2,w^2+xz+y^{3m+3}+y^{2m+2}z(\sum _{i=0}^{m-1}{\alpha }_i{y}^{m-i}+{\alpha }_m)⟩.$$

$$I=⟨xy+z^2,w^2+xz+y^{3m+3}+y^{2m+2}z{\mu }_m⟩.$$

$$⟨xy+z^2,w^2+xz+y^{3m+3}+{ϵ}^2{y}^{2m+2}z{\mu }_{m-1}^{{}^{\prime }}⟩.$$

$$⟨xy+z^2,xz+w^2+y^{3m+3}+y^{2m+2}z⟩.$$

$$I=⟨xy+z^2,w^2+xz+y^{3m+3}+y^{2m+3}z(\sum _{i=0}^{m-2}{\alpha }_i{y}^{m-1-i}+{\alpha }_{m-1})⟩.$$

$$I=⟨xy+z^2,w^2+xz+y^{3m+3}+y^{2m+3}z{\mu }_{m-2}⟩.$$

$$⟨xy+z^2,w^2+xz+y^{3m+3}+{ϵ}^6{y}^{2m+3}z{\mu }_{m-2}^{{}^{\prime }}⟩.$$

$$⟨xy+z^2,xz+w^2+y^{3m+3}+y^{2m+3}z⟩.$$

$$⟨xy+z^2,w^2+xz+y^{3m+3}+y^{3m+2-i}z⟩$$

$$⟨xy+z^2,xz+w^2+z{y}^{2m+2}⟩$$

$$⟨xy+z^2,xz+w^2+z{y}^{2m+2}+y^{4m+4-i}⟩fori=0,1,\dots ,m.$$

$$⟨xy+z^2,w^2+xz+y^{3m+4}⟩$$

$$⟨xy+z^2,w^2+xz+y^{3m+4}+y^{3m+3-i}⟩fori=0,1,\dots ,m.$$

$$⟨xy+z^2,w^2+x^2+y^3⟩$$

$$⟨xy+z^2,w^2+x^2+y^3+y{z}^2⟩.$$

Full text

Mathematics subject classification: Primary 14B05; Secondary 14H20, 14J17.

Reference number #3569

1. Introduction

In this article we report about a classifier for unimodular isolated complete intersection surface singularities in the computer algebra system Singular (cf. [DGPS13],[GP07]). Marc Giusti gave the complete list of simple isolated complete intersection singularities which are not hypersurfaces (cf. [GM83]). Wall achieved the classification of unimodular singularities which are not hypersurfaces (cf. [Wal83]). Two of the authors described Giusti’s classification in terms of certain invariants. Based on this description it is not necessary to compute the normal form for finding the type of the singularity. This is usually more complicated and may be space and time consuming (cf. [ADPG1], [ADPG2]). Similarly the type of singularities in terms of certain invariants (Milnor number, Tjurina number and semi groups) for unimodular isolated complete intersection space curve singularities given by C. T. C. Wall is characterized [ADPG3].

A basis for a classifier is a complete list of these singularities together with a list of invariants characterizing them. Since Wall gave only representatives of the $\mu$-constant strata in his classification (cf. [Wal83]), we complete his list by computing the versal $\mu$-constant deformation of the singularities. The new list obtained in this way contains all unimodular complete intersection surface singularities.

In section 2 we characterize the $2$-jet of the unimodular complete intersection singularities using primary decomposition, Krull dimension and Hilbert polynomials. In section 3 we give the complete list of unimodular complete intersection surface singularities by fixing the $2$-jet of the singularities and develop algorithms for each case. In section 4 we give the main algorithm for computing the type of unimodular complete intersection surface singularity.

2. Characterization of normal form of 2-jet of singularities

Consider $I=\langle f_{1},f_{2}\rangle\subseteq\langle x,y,z,w\rangle^{2}\mathbb{C}[[x,y,z,w]]$ defining a complete intersection singularity and let $I_{2}$ be the $2$-jet of $I$. According to C.T.C. Wall’s classification the $2$-jet of $\langle f_{1},f_{2}\rangle$ is a homogenous ideal generated by 2 polynomials of degree 2. We want to give a description of the type of a singularity without producing the normal form. C.T.C. Wall’s classification is based on the classification of the $2$-jet $I_{2}$ of $\langle f,g\rangle$. Let $\bigcap^{s}_{i=1}Q_{i}$ be the irredundant primary decomposition of $I_{2}$ in $\mathbb{C}[[x,y,z,w]]$. Let $t$ be the number of prime ideals appearing in primary decomposition of $I_{2}$ and $j_{i}$ be the number of conjugates corresponding to each prime ideal. Let $d_{i}=dim_{\mathbb{C}}\mathbb{C}[[x,y,z,w]]/Q_{i},i=1,...,s$ and $h_{i}$ be the Hilbert polynomial of $\mathbb{C}[[x,y,z,w]]/Q_{i}$. According to C.T.C. Wall’s classification we obtain unimodular singularities only in the following cases.

3. Unimodular complete intersection surface singularities

We set

[TABLE]

for brevity.

3.1. I singularities

Assume the $2-jet$ of $<\langle f,g\rangle$ has normal form $\langle xy-xz,yz-xy\rangle$. In this case according to C.T.C.Wall’s classification the unimodular surface singularities with their Milnor number say $\mu$ and Tjurina number $\tau$ are given in the table below.

Proposition 3.1**.**

The unimodular complete intersection surface singularity with Milnor number $\mu=13$ are $I_{1,0}$ with Tjurina number $\tau=13$ defined by the ideal

[TABLE]

and $I_{1,0,1}$ with Tjurina number $\tau=12$ defined by the ideal

[TABLE]

Proof.

In the list of C.T.C. Wall $I_{1,0}$ is the singularity defined by the ideal

[TABLE]

with Milnor number $\mu=13$ and Tjurina number $\tau=13$. The versal deformation of $I_{1,0}$ is given by

[TABLE]

$I_{1,0}$ defines a weighted homogenous singularity with weights, $(w_{1},w_{2},w_{3},w_{4})=(6,6,6,4)$ and the degrees $(d_{1},d_{2})=(12,12)$. The versal $\mu$-constant deformation of $I_{1,0}$ is given by

[TABLE]

Using the coordinate change $x\rightarrow\xi^{6}x$, $y\rightarrow\xi^{6}y$, $z\rightarrow\xi^{6}z$, $w\rightarrow\xi^{4}w$. We obtain

[TABLE]

Choosing $\xi$ such that $\lambda_{1}\xi^{4}=1$. So we obtain

[TABLE]

with Tjurina number $\tau=12$ different from $I_{1,0}$. ∎

Summarizing the results of the above preposition we complete the list of unimodular complete intersection singularities in case of $\langle f,g\rangle$ having 2-jet with normal forms $\langle xy-xz,yz-xy\rangle$.

Proposition 3.2**.**

Let $(V(\langle f,g\rangle),0)\subseteq(\mathbb{C}^{4},0)$ be the germ of a complete intersection surface singularity. Assume it is not a hypersurface singularity and the $2$-jet of $\langle f,g\rangle$ has normal form $\langle xy-xz,yz-xy\rangle$. $(V(\langle f,g\rangle),0)$ is unimodular if and only if it is isomorphic to a complete intersection in Tables 2 and 3.

Proof.

The proof is a direct consequence of C.T.C. Wall’s classification and Propositions 3.1. ∎

We summarize our approach in this case in Algorithm 1

3.2. T singularities

Assume the $2-jet$ of $\langle f,g\rangle$ has normal form $\langle x^{2}+y^{2}+z^{2},y^{2}+z^{2}+w^{2}\rangle$, $\langle xy+z^{2}+w^{2},zw+y^{2}\rangle$, $\langle xy+z^{2}+w^{2},zw\rangle$ or $\langle xy,zw\rangle$. In this case according to C.T.C.Wall’s classification the unimodular surface singularities with Milnor number $\mu$ and Tjurina number $\tau$ are given in the table 4 above.

3.3. $\textbf{J}^{\prime}$ singularities

Assume the $2-jet$ of $\langle f,g\rangle$ has normal form $\langle xy+z^{2},w^{2}+xz\rangle$. According to C.T.C. Wall’s classification the unimodular surface singularities are given in the following table

Proposition 3.3**.**

The unimodular complete intersection surface singularities having Milnor number of the form $\mu=6m+9$ where m is a positive integer are $J^{\prime}_{6m+9}$ with Tjurina number $\tau=6m+9$ defined by the ideal

[TABLE]

and $J^{\prime}_{6m+9,i+1}$ with Tjurina number $\tau=6m+8-i$ are defined by the ideal

[TABLE]

Proof.

In C.T.C. Wall’s list the only unimodular complete intersection surface singularities are the singularities $J^{\prime}_{6m+9}$ defined by the ideal

[TABLE]

The versal deformation of $J^{\prime}_{6m+9}$ is given by

[TABLE]

$J^{\prime}_{6m+9}$ defines a weighted homogenous isolated complete intersection singularity with weights

[TABLE]

and degrees

[TABLE]

The versal $\mu-$constant deformation of $J^{\prime}_{6m+9}$ is

[TABLE]

Consider $\phi\in Aut_{\mathbb{C}}(\mathbb{C}[[x,y,z,w]])$ defined by $\phi(x)=\epsilon^{12m+10}x$, $\phi(y)=\epsilon^{6}y$, $\phi(z)=\epsilon^{6m+8}z$ and $\phi(w)=\epsilon^{9m+9}w$. If $\alpha_{m}\neq 0$, then we can write $I$ as

[TABLE]

Let $\mu_{m-1}=\sum^{m-1}_{i=0}\alpha_{i}y^{m-i}+\alpha_{m}$. Then $I$ can be written

[TABLE]

By applying the transformation $\phi$ we get

[TABLE]

Choosing $\varepsilon$ such that $\varepsilon^{2}\mu^{{}^{\prime}}_{m-1}=1$, we obtain

[TABLE]

Now we assume that $\alpha_{m}=0$. This implies that

[TABLE]

Let $\mu_{m-2}=\sum^{m-2}_{i=0}\alpha_{i}y^{m-1-i}+\alpha_{m-1}$ then we can have

[TABLE]

Now applying the transformation $\phi$ we get $I$ as

[TABLE]

Choosing $\varepsilon$ such that $\varepsilon^{6}\mu^{{}^{\prime}}_{m-2}=1$, so we obtain

[TABLE]

Now we assume $\alpha_{m-1}=0$. Iterating in the same way we get $m+1$ different singularities defined by

[TABLE]

for $i=0,1,\ldots,m$ and $m\geq 1$. ∎

Proposition 3.4**.**

The unimodular complete intersection surface singularities having Milnor number of the form $\mu=6m+10$ where m is a positive integer are $J^{\prime}_{6m+10}$ with Tjurina number $\tau=6m+10$ defined by the ideal

[TABLE]

and $J^{\prime}_{6m+10,i+1}$ with Tjurina numbers $6m+10-i$ is defined by the ideal

[TABLE]

Proposition 3.5**.**

The unimodular complete intersection surface singularities having Milnor number of the form $\mu=6m+11$ where m is a positive integer are $J^{\prime}_{6m+11}$ with Tjurina number $\tau=6m+11$ defined by the ideal

[TABLE]

and $J^{\prime}_{6m+11,i+1}$ with Tjurina number $6m+10-i$ is defined by the ideal

[TABLE]

Proof.

The proofs of Propositions 3.4 and 3.5 can be done similarly to the proof of Proposition 3.3. ∎

Summarizing the above results we have the following table

As a consequence of C.T.C. Wall’s classification and Propositions 3.3 - 3.5 we obtain:

Proposition 3.6**.**

Let $(V(\langle f,g\rangle),0)\subseteq(\mathbb{C}^{4},0)$ be the germ of a complete intersection surface singularity. Assume it is not a hypersurface singularity and the $2$-jet of $\langle f,g\rangle$ has normal form $\langle xy+z2,w^{2}+xz\rangle$. $(V(\langle f,g\rangle),0)$ is unimodular if and only if it is isomorphic to a complete intersection in Tables 5 and 6.

We summarize our approach in this case in Algorithm 3.

3.4. $\textbf{K}^{\prime}$ singularities

Assume the $2-jet$ of $\langle f,g\rangle$ has normal form $\langle xy+z^{2},w^{2}+x^{2}\rangle$. In this case according to C.T.C.Wall’s classification the unimodular surface singularities with Milnor number $\mu$ are given in the table below.

Proposition 3.7**.**

The unimodular complete intersection surface singularities with Milnor number $\mu=10$ are $K^{{}^{\prime}}_{10}$ with Tjurina number $\tau=10$ defined by the ideal

[TABLE]

and $K^{{}^{\prime}}_{10,1}$ with Tjurina number $\tau=9$ defined by the ideal

[TABLE]

Proof.

In the list of C.T.C Wall $K^{{}^{\prime}}_{10}$ is the singularity defined by the ideal

[TABLE]

with Milnor number $\mu=10$ and Tjurina number $\tau=10$. The versal deformation of $K^{{}^{\prime}}_{10}$ is

[TABLE]

$K^{{}^{\prime}}_{10}$ defines a weighted homogenous isolated complete intersection singularity with

[TABLE]

The versal $\mu$-constant deformation of $K^{{}^{\prime}}_{10,1}$ is given by $\langle xy+z^{2},w^{2}+x^{2}+y^{3}+\lambda_{1}yz^{2}\rangle$. Using the coordinate change $x\rightarrow\xi^{6}x$, $y\rightarrow\xi^{4}y$, $z\rightarrow\xi^{5}z$ and $w\rightarrow\xi^{6}w$, we obtain

[TABLE]

Choosing $\xi$ such that $\lambda_{1}\xi^{2}=1$, we obtain

[TABLE]

with Tjurina number $\tau=9$. ∎

Proposition 3.8**.**

The unimodular complete intersection surface singularities with Milnor number $\mu=11$ are $K^{{}^{\prime}}_{11}$ with Tjurina number $\tau=11$ defined by the ideal

[TABLE]

and $K^{{}^{\prime}}_{11,1}$ with Tjurina number $\tau=10$ defined by the ideal

[TABLE]

Proof.

The proof of Proposition 3.8 can be done similarly to the proof of Proposition 3.7. ∎

Proposition 3.9**.**

The unimodular complete intersection surface singularities with Milnor number $\mu=13$ are $K^{{}^{\prime}}_{1,0}$ with Tjurina number $\tau=13$ defined by the ideal

[TABLE]

$K^{{}^{\prime}}_{1,0,1}$* with Tjurina number $\tau=12$ defined by the ideal*

[TABLE]

Proof.

In C.T.C’s Wall list $K^{{}^{\prime}}_{1,0}$ is the singularity defined by the ideal

[TABLE]

with Milnor number $\mu=13$ and Tjurina number $\tau=13.$ The versal deformation of $K^{{}^{\prime}}_{1,0}$ is given by

[TABLE]

$K^{{}^{\prime}}_{1,0}$ defines a weighted homogenous isolated complete intersection singularity with weights

[TABLE]

and the degrees

[TABLE]

The versal $\mu$-constant deformation of $K^{{}^{\prime}}_{1,0}$ is given by

[TABLE]

Using the coordinate change $x\rightarrow\xi^{4}x$, $y\rightarrow\xi^{2}y$, $z\rightarrow\xi^{3}z$,$w\rightarrow\xi^{4}w$, we obtain

[TABLE]

Choosing $\xi$ such that $\lambda_{5}\xi^{2}=1$ we obtain

[TABLE]

with Tjurina number $\tau=12$. ∎

Proposition 3.10**.**

The unimodular complete intersection surface singularities with Milnor number $\mu=15$ are $K^{{}^{\prime}}_{15}$ with Tjurina number $\tau=15$ defined by the ideal

[TABLE]

and $K^{{}^{\prime}}_{15,1}$ with Tjurina number $\tau=14$ defined by the ideal

[TABLE]

and $K^{{}^{\prime}}_{15,2}$ with Tjurina number $\tau=13$ defined by the ideal

[TABLE]

Proof.

In C.T.C. Wall’s list $K^{{}^{\prime}}_{15}$ is the singularity defined by the ideal

[TABLE]

with Milnor number $\mu=15$ and Tjurina number $\tau=15$. The versal deformation of $K^{{}^{\prime}}_{15}$ is given by

[TABLE]

$K^{{}^{\prime}}_{15}$ defines a weighted homogenous isolated complete intersection singularity with weights

[TABLE]

and the degrees

[TABLE]

The versal $\mu$-constant deformation of $K^{{}^{\prime}}_{15}$ is given by

[TABLE]

If $\lambda_{6}\neq 0$ then

[TABLE]

Using coordinate change such that $x\rightarrow\xi^{7}x$, $y\rightarrow\xi^{3}y$, $z\rightarrow\xi^{5}z$ and $w\rightarrow\xi^{7}w$ we may assume

[TABLE]

Choosing $\xi$ such that $\xi\bar{u}=1$ we obtain

[TABLE]

with Tjurina number $\tau=13$. If $\lambda_{6}=0$ then again applying the same coordinate change we obtain

[TABLE]

by choosing $\lambda_{5}\xi^{4}=1$ with Tjurina number $\tau=14$. ∎

Proposition 3.11**.**

The unimodular complete intersection surface singularities with Milnor number $\mu=16$ are $K^{{}^{\prime}}_{16}$ with Tjurina number $\tau=16$ defined by the ideal

[TABLE]

$K^{{}^{\prime}}_{16,1}$* with Tjurina number $\tau=15$ defined by the ideal*

[TABLE]

and $K^{{}^{\prime}}_{16,2}$ with Tjurina number $\tau=14$ defined by the ideal

[TABLE]

Proof.

The proof of Proposition 3.11 can be done similarly to the proof of Proposition 3.10. ∎

Summarizing the results of the above propositions we complete the list of unimodular complete intersection singularities in case of $\langle f,g\rangle$ having 2-jet with normal forms $\langle xy+z^{2},w^{2}+x^{2}\rangle$.

Proposition 3.12**.**

Let $(V(\langle f,g\rangle),0)\subseteq(\mathbb{C}^{4},0)$ be the germ of a complete intersection surface singularity. Assume it is not a hypersurface singularity and the $2$-jet of $\langle f,g\rangle$ has normal form $\langle xy+z^{2},w^{2}+x^{2}\rangle$. $(V(\langle f,g\rangle),0)$ is unimodular if and only if it is isomorphic to a complete intersection in Tables 7 and 8.

Proof.

The proof is a direct consequence of C.T.C. Wall’s classification and Propositions 3.7 - 3.11 ∎

Proposition 3.13**.**

Let $(\mathbf{V}(\langle f,g\rangle),0)\subseteq(\mathbb{C}^{4},0)$ be the germ of a complete intersection surface singularity. Assume it is not hypersurface singularity and the two jet of $\langle f,g\rangle$ has normal form $\langle xy+z^{2},w^{2}+xz\rangle$. $\mathbf{V}(\langle f,g\rangle,0)$ is unimodular if and only if it is isomorphic to a complete intersection in table $3$.

We summarize our approach in this case in Algorithm 4.

3.5. L singularities

Assume the $2-jet$ of $\langle f,g\rangle$ has normal form $\langle wz+xy,y^{2}+xz\rangle$. According to C.T.C. Wall’s classification the unimodular singularities are given in the table.

Proposition 3.14**.**

The unimodular complete intersection surface singularity with Milnor number $\mu=10$ and Tjurina number $\tau=10$ is $L_{10}$ defined by the ideal

[TABLE]

and $L_{10,1}$ with Tjurina number $\tau=9$ is defined by the ideal

[TABLE]

Proof.

In C.T.C. Wall’s list the only unimodular complete intersection singularity with Milnor number $\mu=10$ is the singularity $L_{10}$ defined by the ideal $(wz+xy,y^{2}+xz+w^{3})$. The versal deformation of the above singularity is given by

[TABLE]

$L_{10}$ defines a weighted homogenous isolated complete intersection singularity with weights

[TABLE]

and degrees

[TABLE]

The versal $\mu-$constant deformation of $L_{10}$ is

[TABLE]

Using the coordinate change

[TABLE]

we obtain

[TABLE]

Take $\varepsilon$ such that $\varepsilon^{2}A^{{}^{\prime}}=1$ then we obtain

[TABLE]

∎

Proposition 3.15**.**

The unimodular complete intersection surface singularity with Milnor number $\mu=11$ and Tjurina number $\tau=11$ is $L_{11}$ defined by the ideal

[TABLE]

and $L_{11,1}$ with Tjurina number $\tau=10$ is defined by the ideal

[TABLE]

Proof.

The proof of Proposition 3.15 can be done similarly to the proof of Proposition 3.14. ∎

Proposition 3.16**.**

The unimodular complete intersection surface singularity with Milnor number $\mu=13$ and with Tjurina number $\tau=13$ is $L_{1,0}$ defined by the ideal

[TABLE]

and $L_{1,0,1}$ with Tjurina number $\tau=12$ is defined by the ideal

[TABLE]

Proof.

In C.T.C. Wall’s list the only unimodular complete intersection singularity with Milnor number $13$ is the singularity $L_{1,0}$ defined by the ideal

[TABLE]

The versal deformation of the above singularity is given by

[TABLE]

$L_{1,0}$ defines a weighted homogenous isolated complete intersection singularity with weights

[TABLE]

and degrees

[TABLE]

The versal $\mu-$constant deformation of $L_{1,0}$ is

[TABLE]

Now using the coordinate change $x\rightarrow\varepsilon^{3}x$, $y\rightarrow\varepsilon^{4}y$, $z\rightarrow\varepsilon^{5}z$ and $w\rightarrow\varepsilon^{2}w$. We obtain

[TABLE]

Choosing $\varepsilon$ such that $\varepsilon^{2}A^{{}^{\prime}}=1$. We obtain

[TABLE]

with Tjurina number $\tau=9$. ∎

Proposition 3.17**.**

The unimodular complete intersection surface singularity with Milnor number $\mu=15$ and with Tjurina number $\tau=15$ is $L_{15}$ defined by the ideal

[TABLE]

$L_{15,1}$* with Tjurina number $\tau=14$ is defined by the ideal*

[TABLE]

and $L_{15,2}$ with Tjurina number $\tau=13$ is defined by the ideal

[TABLE]

Proof.

In C.T.C. Wall’s list the only unimodular complete intersection singularity with Milnor number $\mu=15$ is the singularity $L_{15}$ defined by the ideal

[TABLE]

The versal deformation of the above singularity is given by

[TABLE]

$L_{15}$ defines a weighted homogenous isolated complete intersection singularity with weights

[TABLE]

and degrees

[TABLE]

The versal $\mu-$constant deformation of $L_{15}$ is

[TABLE]

Let $B\neq 0$,

[TABLE]

Let $\mu_{B}=Aw+B$, Now using the coordinate change $x\rightarrow\varepsilon^{5}x$, $y\rightarrow\varepsilon^{7}y$, $z\rightarrow\varepsilon^{9}z$ and $w\rightarrow\varepsilon^{3}w$, we obtain

[TABLE]

Choosing $\varepsilon$ such that $\varepsilon\acute{\mu_{B}}=1$. So we obtain

[TABLE]

with Tjurina number $\tau=13$. Now we if $B=0$ then it becomes

[TABLE]

Now using again the same coordinate change we obtain

[TABLE]

Choosing $\varepsilon$ such that $\varepsilon^{4}A^{{}^{\prime}}=1$ so we obtain

[TABLE]

with Tjurina number $\tau=14$. ∎

Proposition 3.18**.**

The unimodular complete intersection surface singularity with Milnor number $\mu=16$ and Tjurina number $\tau=16$ is $L_{16}$ defined by the ideal

[TABLE]

$L_{16,1}$* with Tjurina number $\tau=15$ is defined by the ideal*

[TABLE]

and and $L_{16,2}$ with Tjurina number $\tau=14$ is defined by the ideal

[TABLE]

Summarizing the above results we have the following table.

Proposition 3.19**.**

Let $(V(\langle f,g\rangle),0)\subseteq(\mathbb{C}^{4},0)$ be the germ of a complete intersection surface singularity. Assume it is not a hypersurface singularity and the $2$-jet of $\langle f,g\rangle$ has normal form $\langle wz+xy,y^{2}+xz\rangle$. $(V(\langle f,g\rangle),0)$ is unimodular if and only if it is isomorphic to a complete intersection in Tables 9 and 10.

3.6. M singularities

Assume the $2-jet$ of $\langle f,g\rangle$ has normal form $\langle wy+x^{2}-z^{2},wx\rangle$. In this case According to C.T.C.Wall’s classification the unimodular surface singularities with Milnor number $\mu$ are given in the table below.

Proposition 3.20**.**

The unimodular complete intersection surface singularities with Milnor number $\mu=11$ are $M_{11}$ with Tjurina number $\tau=11$ defined by the ideal

[TABLE]

and $M_{11,1}$ with Tjurina number $\tau=10$ defined by the ideal

[TABLE]

Proof.

In C.T.C. Wall’s list $M_{11}$ is the singularity defined by the ideal

[TABLE]

with Milnor number $\mu=11$ and Tjurina number $\tau=11$. The versal deformation of $M_{11}$ is

[TABLE]

$M_{11}$ defines a weighted homogenous isolated complete intersection singularity with weights

[TABLE]

and the degrees

[TABLE]

The versal $\mu$-constant deformation of $M_{11}$ is given by

[TABLE]

Using the coordinate change $x\rightarrow\xi^{4}x$, $y\rightarrow\xi^{3}y$, $z\rightarrow\xi^{4}z$ and $w\rightarrow\xi^{5}w$ we obtain

[TABLE]

Choosing $\xi$ such that $\lambda_{1}\xi=1$ we obtain

[TABLE]

with different Tjurina number $\tau=10$ from $M_{11}$. ∎

Proposition 3.21**.**

The unimodular complete intersection surface singularities with Milnor number $\mu=13$ are $M_{1,0}$ with Tjurina number $\tau=13$ defined by the ideal

[TABLE]

$M_{1,0,1}$* with Tjurina number $\tau=12$ defined by the ideal*

[TABLE]

Proof.

In the list of C.T.C Wall $M_{1,0}$ is the singularity defined by the ideal

[TABLE]

with Milnor number $\mu=13$ and Tjurina number $\tau=13$. The versal deformation of $M_{1,0}$ is

[TABLE]

$M_{1,0}$ defines a weighted homogenous isolated complete intersection singularity with weights

[TABLE]

and the degrees

[TABLE]

The versal $\mu$-constant deformation of $M_{1,0}$ is given by

[TABLE]

Using the coordinate change $x\rightarrow\xi^{6}x$, $y\rightarrow\xi^{4}y$, $z\rightarrow\xi^{6}z$, $w\rightarrow\xi^{8}w$ we obtain

[TABLE]

Choosing $\xi$ such that $\lambda_{3}\xi^{4}=1$

[TABLE]

with Tjurina number $\tau=12$. ∎

Proposition 3.22**.**

The unimodular complete intersection surface singularities with Milnor number $\mu=15$ are $M_{15}$ with Tjurina number $\tau=15$ defined by the ideal

[TABLE]

$M_{15,1}$* with Tjurina number $\tau=14$ defined by the ideal*

[TABLE]

and $M_{15,2}$ with Tjurina number $\tau=13$ defined by the ideal

[TABLE]

Proof.

In the list of C.T.C Wall $M_{15}$ is the singularity defined by the ideal

[TABLE]

with Milnor number $\mu=15$ and Tjurina number $\tau=15.$ The versal deformation of $M_{15}$ is

[TABLE]

$M_{15}$ defines a weighted homogenous isolated complete intersection singularity with weights

[TABLE]

and the degrees

[TABLE]

The versal $\mu$-constant deformation of $M_{15}$ is given by

[TABLE]

If $\lambda_{2}\neq 0$ then we have

[TABLE]

where $u=\lambda_{1}y+\lambda_{2}$. Using the coordinate change $x\rightarrow\xi^{6}x$, $y\rightarrow\xi^{4}y$, $z\rightarrow\xi^{6}z$ and $w\rightarrow\xi^{8}w$ we obtain

[TABLE]

Choosing $u\xi^{2}=1$ we obtain

[TABLE]

with Tjurina number $\tau=14$. If $\lambda_{2}=0$ then again by the same transformation we obtain

[TABLE]

with Tjurina number $\tau=13$ by choosing $\lambda_{1}\xi^{2}=1$. ∎

Summarizing the results of the above prepositions we complete the list of unimodular complete intersection singularities in case of $\langle f,g\rangle$ having 2-jet with normal forms $\langle wy+x^{2}-z^{2},wx\rangle$.

Proposition 3.23**.**

Let $(V(\langle f,g\rangle),0)\subseteq(\mathbb{C}^{4},0)$ be the germ of a complete intersection surface singularity. Assume it is not a hypersurface singularity and the $2$-jet of $\langle f,g\rangle$ has normal form $\langle wy+x^{2}-z^{2},wx\rangle$. $(V(\langle f,g\rangle),0)$ is unimodular if and only if it is isomorphic to a complete intersection in Tables 11 and 12.

Proof.

The proof is a direct consequence of C.T.C. Wall’s classification and Propositions 3.20 - 3.22 ∎

We summarize our approach in this case in Algorithm 6.

4. Main Algorithm

Now we give our main Algorithm 7 by using the all Algorithms given in section 3 by which we can compute the type of the singularity.

Acknowledgements The part of this work is carried out at Kaiserslautern University, Germany. We are thankful to DAAD Germany for the financial support.