Unimodular ICIS, a Classifier
Deeba Afzal, Farkhanda Afzal, Sidra Mubarak, Gerhard Pfister, Asad, Yaqub

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 -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.
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Computational Geometry and Mesh Generation · Polynomial and algebraic computation
Unimodular ICIS, A CLASSIFIER
Deeba Afzal
Deeba Afzal
Department of Mathematics
University of Kaiserslautern
Erwin-Schrödinger-Str.
67663 Kaiserslautern
Germany.
Department of Mathematics
University of Lahore
Near Raiwind Road
Lahore 54000
Pakistan
,
Farkhanda Afzal
Farkhanda Afzal
School of Mathematics and systems science
Beihang University
SMSS, 37 Xueyuan Lu, Haitian District
Beijing 100096
China
,
Sidra Mubarak
Sidra Mubarak
Department of Mathematics
University of Lahore
Near Raiwind Road
Lahore 54000
Pakistan
,
Gerhard Pfister
Gerhard Pfister
Department of Mathematics
University of Kaiserslautern
Erwin-Schrödinger-Str.
67663 Kaiserslautern
Germany
and
Asad Yaqub
Asad Yaqub
Department of Mathematics
University of Lahore
Near Raiwind Road
Lahore 54000
Pakistan
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 -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 .
Key words and phrases:
Singularities, Milnor number, Tjurina number, blowing up.
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 -constant strata in his classification (cf. [Wal83]), we complete his list by computing the versal -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 -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 -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 defining a complete intersection singularity and let be the -jet of . According to C.T.C. Wall’s classification the -jet of 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 -jet of . Let be the irredundant primary decomposition of in . Let be the number of prime ideals appearing in primary decomposition of and be the number of conjugates corresponding to each prime ideal. Let and be the Hilbert polynomial of . 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 of has normal form . In this case according to C.T.C.Wall’s classification the unimodular surface singularities with their Milnor number say and Tjurina number are given in the table below.
Proposition 3.1**.**
The unimodular complete intersection surface singularity with Milnor number are with Tjurina number defined by the ideal
[TABLE]
and with Tjurina number defined by the ideal
[TABLE]
Proof.
In the list of C.T.C. Wall is the singularity defined by the ideal
[TABLE]
with Milnor number and Tjurina number . The versal deformation of is given by
[TABLE]
defines a weighted homogenous singularity with weights, and the degrees . The versal -constant deformation of is given by
[TABLE]
Using the coordinate change , , , . We obtain
[TABLE]
Choosing such that . So we obtain
[TABLE]
with Tjurina number different from . ∎
Summarizing the results of the above preposition we complete the list of unimodular complete intersection singularities in case of having 2-jet with normal forms .
Proposition 3.2**.**
Let be the germ of a complete intersection surface singularity. Assume it is not a hypersurface singularity and the -jet of has normal form . 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 of has normal form , , or . In this case according to C.T.C.Wall’s classification the unimodular surface singularities with Milnor number and Tjurina number are given in the table 4 above.
3.3. singularities
Assume the of has normal form . 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 where m is a positive integer are with Tjurina number defined by the ideal
[TABLE]
and with Tjurina number are defined by the ideal
[TABLE]
Proof.
In C.T.C. Wall’s list the only unimodular complete intersection surface singularities are the singularities defined by the ideal
[TABLE]
The versal deformation of is given by
[TABLE]
defines a weighted homogenous isolated complete intersection singularity with weights
[TABLE]
and degrees
[TABLE]
The versal constant deformation of is
[TABLE]
Consider defined by , , and . If , then we can write as
[TABLE]
Let . Then can be written
[TABLE]
By applying the transformation we get
[TABLE]
Choosing such that , we obtain
[TABLE]
Now we assume that . This implies that
[TABLE]
Let then we can have
[TABLE]
Now applying the transformation we get as
[TABLE]
Choosing such that , so we obtain
[TABLE]
Now we assume . Iterating in the same way we get different singularities defined by
[TABLE]
for and . ∎
Proposition 3.4**.**
The unimodular complete intersection surface singularities having Milnor number of the form where m is a positive integer are with Tjurina number defined by the ideal
[TABLE]
and with Tjurina numbers is defined by the ideal
[TABLE]
Proposition 3.5**.**
The unimodular complete intersection surface singularities having Milnor number of the form where m is a positive integer are with Tjurina number defined by the ideal
[TABLE]
and with Tjurina number 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 be the germ of a complete intersection surface singularity. Assume it is not a hypersurface singularity and the -jet of has normal form . 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. singularities
Assume the of has normal form . In this case according to C.T.C.Wall’s classification the unimodular surface singularities with Milnor number are given in the table below.
Proposition 3.7**.**
The unimodular complete intersection surface singularities with Milnor number are with Tjurina number defined by the ideal
[TABLE]
and with Tjurina number defined by the ideal
[TABLE]
Proof.
In the list of C.T.C Wall is the singularity defined by the ideal
[TABLE]
with Milnor number and Tjurina number . The versal deformation of is
[TABLE]
defines a weighted homogenous isolated complete intersection singularity with
[TABLE]
The versal -constant deformation of is given by . Using the coordinate change , , and , we obtain
[TABLE]
Choosing such that , we obtain
[TABLE]
with Tjurina number . ∎
Proposition 3.8**.**
The unimodular complete intersection surface singularities with Milnor number are with Tjurina number defined by the ideal
[TABLE]
and with Tjurina number 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 are with Tjurina number defined by the ideal
[TABLE]
* with Tjurina number defined by the ideal*
[TABLE]
Proof.
In C.T.C’s Wall list is the singularity defined by the ideal
[TABLE]
with Milnor number and Tjurina number The versal deformation of is given by
[TABLE]
defines a weighted homogenous isolated complete intersection singularity with weights
[TABLE]
and the degrees
[TABLE]
The versal -constant deformation of is given by
[TABLE]
Using the coordinate change , , ,, we obtain
[TABLE]
Choosing such that we obtain
[TABLE]
with Tjurina number . ∎
Proposition 3.10**.**
The unimodular complete intersection surface singularities with Milnor number are with Tjurina number defined by the ideal
[TABLE]
and with Tjurina number defined by the ideal
[TABLE]
and with Tjurina number defined by the ideal
[TABLE]
Proof.
In C.T.C. Wall’s list is the singularity defined by the ideal
[TABLE]
with Milnor number and Tjurina number . The versal deformation of is given by
[TABLE]
defines a weighted homogenous isolated complete intersection singularity with weights
[TABLE]
and the degrees
[TABLE]
The versal -constant deformation of is given by
[TABLE]
If then
[TABLE]
Using coordinate change such that , , and we may assume
[TABLE]
Choosing such that we obtain
[TABLE]
with Tjurina number . If then again applying the same coordinate change we obtain
[TABLE]
by choosing with Tjurina number . ∎
Proposition 3.11**.**
The unimodular complete intersection surface singularities with Milnor number are with Tjurina number defined by the ideal
[TABLE]
* with Tjurina number defined by the ideal*
[TABLE]
and with Tjurina number 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 having 2-jet with normal forms .
Proposition 3.12**.**
Let be the germ of a complete intersection surface singularity. Assume it is not a hypersurface singularity and the -jet of has normal form . 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 be the germ of a complete intersection surface singularity. Assume it is not hypersurface singularity and the two jet of has normal form . is unimodular if and only if it is isomorphic to a complete intersection in table .
We summarize our approach in this case in Algorithm 4.
3.5. L singularities
Assume the of has normal form . 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 and Tjurina number is defined by the ideal
[TABLE]
and with Tjurina number is defined by the ideal
[TABLE]
Proof.
In C.T.C. Wall’s list the only unimodular complete intersection singularity with Milnor number is the singularity defined by the ideal . The versal deformation of the above singularity is given by
[TABLE]
defines a weighted homogenous isolated complete intersection singularity with weights
[TABLE]
and degrees
[TABLE]
The versal constant deformation of is
[TABLE]
Using the coordinate change
[TABLE]
we obtain
[TABLE]
Take such that then we obtain
[TABLE]
∎
Proposition 3.15**.**
The unimodular complete intersection surface singularity with Milnor number and Tjurina number is defined by the ideal
[TABLE]
and with Tjurina number 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 and with Tjurina number is defined by the ideal
[TABLE]
and with Tjurina number is defined by the ideal
[TABLE]
Proof.
In C.T.C. Wall’s list the only unimodular complete intersection singularity with Milnor number is the singularity defined by the ideal
[TABLE]
The versal deformation of the above singularity is given by
[TABLE]
defines a weighted homogenous isolated complete intersection singularity with weights
[TABLE]
and degrees
[TABLE]
The versal constant deformation of is
[TABLE]
Now using the coordinate change , , and . We obtain
[TABLE]
Choosing such that . We obtain
[TABLE]
with Tjurina number . ∎
Proposition 3.17**.**
The unimodular complete intersection surface singularity with Milnor number and with Tjurina number is defined by the ideal
[TABLE]
* with Tjurina number is defined by the ideal*
[TABLE]
and with Tjurina number is defined by the ideal
[TABLE]
Proof.
In C.T.C. Wall’s list the only unimodular complete intersection singularity with Milnor number is the singularity defined by the ideal
[TABLE]
The versal deformation of the above singularity is given by
[TABLE]
defines a weighted homogenous isolated complete intersection singularity with weights
[TABLE]
and degrees
[TABLE]
The versal constant deformation of is
[TABLE]
Let ,
[TABLE]
Let , Now using the coordinate change , , and , we obtain
[TABLE]
Choosing such that . So we obtain
[TABLE]
with Tjurina number . Now we if then it becomes
[TABLE]
Now using again the same coordinate change we obtain
[TABLE]
Choosing such that so we obtain
[TABLE]
with Tjurina number . ∎
Proposition 3.18**.**
The unimodular complete intersection surface singularity with Milnor number and Tjurina number is defined by the ideal
[TABLE]
* with Tjurina number is defined by the ideal*
[TABLE]
and and with Tjurina number is defined by the ideal
[TABLE]
Summarizing the above results we have the following table.
Proposition 3.19**.**
Let be the germ of a complete intersection surface singularity. Assume it is not a hypersurface singularity and the -jet of has normal form . is unimodular if and only if it is isomorphic to a complete intersection in Tables 9 and 10.
3.6. M singularities
Assume the of has normal form . In this case According to C.T.C.Wall’s classification the unimodular surface singularities with Milnor number are given in the table below.
Proposition 3.20**.**
The unimodular complete intersection surface singularities with Milnor number are with Tjurina number defined by the ideal
[TABLE]
and with Tjurina number defined by the ideal
[TABLE]
Proof.
In C.T.C. Wall’s list is the singularity defined by the ideal
[TABLE]
with Milnor number and Tjurina number . The versal deformation of is
[TABLE]
defines a weighted homogenous isolated complete intersection singularity with weights
[TABLE]
and the degrees
[TABLE]
The versal -constant deformation of is given by
[TABLE]
Using the coordinate change , , and we obtain
[TABLE]
Choosing such that we obtain
[TABLE]
with different Tjurina number from . ∎
Proposition 3.21**.**
The unimodular complete intersection surface singularities with Milnor number are with Tjurina number defined by the ideal
[TABLE]
* with Tjurina number defined by the ideal*
[TABLE]
Proof.
In the list of C.T.C Wall is the singularity defined by the ideal
[TABLE]
with Milnor number and Tjurina number . The versal deformation of is
[TABLE]
defines a weighted homogenous isolated complete intersection singularity with weights
[TABLE]
and the degrees
[TABLE]
The versal -constant deformation of is given by
[TABLE]
Using the coordinate change , , , we obtain
[TABLE]
Choosing such that
[TABLE]
with Tjurina number . ∎
Proposition 3.22**.**
The unimodular complete intersection surface singularities with Milnor number are with Tjurina number defined by the ideal
[TABLE]
* with Tjurina number defined by the ideal*
[TABLE]
and with Tjurina number defined by the ideal
[TABLE]
Proof.
In the list of C.T.C Wall is the singularity defined by the ideal
[TABLE]
with Milnor number and Tjurina number The versal deformation of is
[TABLE]
defines a weighted homogenous isolated complete intersection singularity with weights
[TABLE]
and the degrees
[TABLE]
The versal -constant deformation of is given by
[TABLE]
If then we have
[TABLE]
where . Using the coordinate change , , and we obtain
[TABLE]
Choosing we obtain
[TABLE]
with Tjurina number . If then again by the same transformation we obtain
[TABLE]
with Tjurina number by choosing . ∎
Summarizing the results of the above prepositions we complete the list of unimodular complete intersection singularities in case of having 2-jet with normal forms .
Proposition 3.23**.**
Let be the germ of a complete intersection surface singularity. Assume it is not a hypersurface singularity and the -jet of has normal form . 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.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[ADPG 2] Afzal, D.; Pfister, G.: classifyci.lib . A Singular 3-1-6 library for classifying simple isolated complete singularities for the base field of characteristic 0 (2013).
- 3[ADPG 3] Afzal, D.; Pfister, G.: A classifier for unimodular isolated complete interssection curve singularities . Analele Univ. Ovidius din Constanta, Math. Series 24(1), (2016), 95-119.
- 4[JP 00] De Jong, T.; Pfister, G.: Local Analytic Geometry . Vieweg (2000).
- 5[DGPS 13] Decker, W.; Greuel, G.-M.; Pfister, G.; Schönemann, H.: Singular 3-1-6 – A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2013).
- 6[GM 83] Giusti, M.: Classification des Singularités Isolées Simples d’Intersections Complètes . Proc. Symp. Pure Math. 40, (1983), 457-494.
- 7[GM 75] Greuel, G.-M.: Der Gauß-Manin-Zusamnenhang isolierter singul a ¨ ¨ 𝑎 \ddot{a} ritaten von vollst a ¨ ¨ 𝑎 \ddot{a} ndigen Durchschnitten . Math. Ann. 214(1975), 235-266.
- 8[GP 07] Greuel, G.-M.: Pfister, G.: A Singular Introduction to Commutative Algebra. Second edition, Springer (2007).
