DOI: 10.1515/auom-2016-0005
An. Şt. Univ. Ovidius Constanţa
Vol. 24(1),2016, 95–119
A Classifier for Unimodular Isolated Complete
Intersection Space Curve Singularities
Deeba Afzal and Gerhard Pfister
Abstract
C.T.C. Wall classified the unimodular complete intersection singularities. He indicated in the list only the µ-constant strata and not the
complete classification in each case. In this article we give a complete
list of space curve unimodular singularities and also the description of
a classifier. Instead of computing the normal forms, the singularity is
identified by certain invariants.
1
Introduction
Marc Giusti gave the complete list of simple isolated complete intersection
singularities which are not hypersurfaces (cf. [GM83]). An implementation
in Singular for the classification of simple isolated complete intersection
singularities over the complex numbers is given by Gerhard Pfister and Deeba
Afzal in classifyci.lib as a Singular library (cf. [ADPG1], ADPG2). Wall
achieved the classification of contact unimodular singularities which are not
hypersurfaces (cf. [Wal83]).
We report about a classifier for unimodular isolated complete intersection
curve singularities in the computer algebra system Singular
(cf. [DGPS13],[GP07]). 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 µ-constant strata in his classification (cf. [Wal83]),
we complete his list by computing the versal µ-constant deformation of the
Key Words: Milnor number, Tjurina number, semigroup, versal deformation.
2010 Mathematics Subject Classification: Primary 14B05; Secondary 14H20, 14J17.
Received: 17.03.2014
Accepted: 13.08.2014.
95
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singularities. The new list obtained in this way contains all unimodular complete intersection curve singularities. In section 2 we characterize the 2-jet of
the unimodular complete intersection singularities by using primary decomposition and Hilbert polynomials. In section 3 we give the complete list of
unimodular complete intersection space curve singularities by fixing the 2-jet
of the singularities and develop algorithms for each case. In section 4 we
present examples.
Let us recall the basic definitions.
Let C[[x]] = C[[x1 , ..., xn ]] be the local ring of formal power series and
hxi = hx1 , ..., xn i its maximal ideal.
Definition 1.1. f =< f1 , f2 , ..., fp >, is called complete intersection if
dim C[[x]]/hf1 , . . . , fi i = n − i ,∀ i = 1, . . . , p.
Hypersurfaces are special cases of complete intersections for p = 1.
Definition 1.2. Let f = hf1 , . . . , fp i ⊆ C[[x]] be a complete intersection.
f = hf1 , . . . , fp i has an isolated singularity at 0, if
∂fi
).
1. hf1 , . . . , fp , M1 , . . . Mk i ⊆ hxi, M1 , . . . , Mk the p × p–minors of ( ∂x
j
2. hxic ⊆ hf1 , . . . , fp , M1 , . . . Mk i for some c > 0.
The Milnor number µ(f ) is defined as follows
µ(f ) =
p
X
(−1)p−i dimC C[[x]]/Ci
i=1
1 ,...,fi )
with Ci = hf1 , f2 , ..., fi−1 , ∂(f
∂xj1 ...xji , 1 ≤ j1 , . . . < ji ≤ ni (cf. [GM75]).
The Tjurina number of f is defined to be
dimC C[[x]]p /f C[[x]]p +
n
X
∂f
C[[x]].
∂xi
i=1
Let In,p be the set of all isolated complete intersection singularities. Then
Gc = Aut(C[[x]]) × GLp (C[[x]]) acts on In,p as follows:
let (φ, ψ) ∈ Gc and f = (f1 , . . . , fp ) ∈ In,p then (φ, ψ)(f ) = ψ −1 ◦ f ◦ φ.
Definition 1.3. Let f and g ∈ In,p are called contact equivalent, if there
exists (φ, ψ) ∈ Gc , such that f = (φ, ψ)(g).
In,p ⊆ C[[x]]p caries a canonical topology. It is the topology such that the
maps
C[[x]]p → (C[[x]]/hxic )p
are continuous ∀ c. Here we consider the classical topology of the affine space
(C[[x]]/hxic )p .
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Definition 1.4. An element f ∈ In,p is called simple singularity, if there
exists a neighborhood of f in In,p containing only finitely many orbits of G.
In other words the modality of the singularity is zero.
Definition 1.5. f is called unimodular singularity if there exists a neighborhood of f in In,p containing only one–dimensional families of orbits of Gc . In
other words the modality of the singularity is 1.
Definition 1.6. f ∈ In,p defines a curve if C[[x]]/f is of of dimension 1.
If f defines an irreducible curve , i:e f ⊆ C[[x]] is prime. Then the normalization of C[[x]]/f is C[[t]] and we have parametrization
C[[x]]/f ∼
= K[[x1 (t), x2 (t), ..., xn (t)]] ⊆ C[[t]]
Definition 1.7.
Γf = {ord(f ) | f ∈ C[[x1 (t), x2 (t), ..., xn (t)]]}
is the semi group of the curve.
Definition 1.8. f = hf1 , . . . , fp i ∈ In,p then F = hF1 , . . . , Fp i,
where t = {t1 , . . . , tn } is a deformation of f if
Fi ∈ C[[x, t]]
C[[x]]/hf i ∼
= C[[x]]/hF (x, 0)i.
Any deformation can be induced from the versal deformation by specifying
parameters. P
F = f + ti mi is a versal deformation of f where m1 , . . . , mτ is basis for




∂f1 /∂xn
∂f1 /∂x1




..
..
C[[x]]p /f C[[x]]p + 
 C[[x]].
 C[[x]] + . . . + 
.
.
∂fp /∂x1
2
∂fp /∂xn
Characterization of normal form
of 2-jet of singularities
Let I = hf, gi ⊆ hx, y, zi2 C[[x, y, z]] defines a complete intersection singularity
and I2 be the 2-jet of I. According to C.T.C. Wall’s classification the 2jet of hf, gi 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
Ts
the 2-jet I2 of hf, gi. Let i=1 Qi be the irredundant primary decomposition
of I2 in C[x, y, z]. Let di = dimC (C[x, y, z]/Qi ), i = 1, ..., s and hi be the
Hilbert polynomial of C[x, y, z]/Qi . According to C.T.C. Wall’s classification
we obtain unimodular singularities only in the following cases.
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Table 1:
Type
P
J
F
H
G
K
3
Characterization
s = 2, d1 = d2 = 1 h1 = h2 = 2
s = 2, d1 = d2 = 1 h1 = 1, h2 = 4
s = 2, d1 = 1, d2 = 2 h1 = 1, h2 = 1 + t
s = 2, d1 = 1, d2 = 2 h1 = 2, h2 = 1 + t
√ 3
s = 1 and I2 ⊆ I2
√ 3
s = 1 and I2 6⊆ I2
Normal form of I2
hx2 , yzi
hxy + z 2 , xzi
hxy, xzi
hxy, x2 i
hx2 , y 2 i
hxy + z 2 , x2 i
Unimodular complete intersection space curve singularities
We set
(
xy q ,
li (x, y) =
y q+2 ,
if i = 2q
if i = 2q + 1
for brevity.
Assume the 2-jet of hf, gi has normal form hxy, z 2 i. In this case according
to C.T.C. Wall’s classification the unimodular space curves are given in the
table below
Table 2:
Type
Normal Form
µ
τ
Semigroup
Pk,l
hxy, xk + y l + z 2 i
k ≥ l ≥ 3, k > 3
l+k+1
l+k+1
h1i, h1i, h1i, h1i, h1i l, k even
h1i, h1i, h2, ki l is even, k is odd
h2, ki, h2, li l, k odd
Proposition 3.1. Let I =< f, g >⊆ C[[x, y, z]] defines an isolated complete
intersection singularity (V (I), 0) ⊆ (C3 , 0). Let µ be the Milnor number of I.
Assume that the 2-jet of I has normal form hx2 , yzi.
If (V (I), 0) has 4 branches and all branches are smooth, let J1 , J2 , J3 be
the ideals of the strict transform of I blowing up C3 at 0 corresponding to the
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three affine charts. Assume (V (J1 ), 0) is an Al−3 singularity and (V (J2 ), 0)
is an Ak−3 singularity. Then I is unimodular of type Pk,l , k ≥ 4 and l ≥ 3.
If (V (I), 0) has 3 branches, two branches are smooth and the third branch
has a semigroup generated by (2, k) then I is unimodular of type Pk,µ−k−1 if
(k, µ) 6= (4, 8) and µ − k > 3.
If (V (I), 0) has 2 branches and the semigroup of the two branches are (2, k)
and (2, l) then I is unimodular of type Pk,l if (k, l) 6= (3, 3) and (k, l) 6= (5, 3).
Proof. Using lemma 3.2 (cf. [ADPG1]) we may assume I = hx2 + y k + z l +
g, yz + hi, 3 ≤ k ≤ l ≤ ∞, g ∈ hx, y, zil+1 , h ∈ hx, y, zi3 . According to
Wall’s classification
we may assume that I = hyz, x2 + z l + y k i. Then I =
T
2
k
2
hz, x + y i hy, x + z l i. If l and k are even then (V (I), 0) has 4 smooth
branches. If l + k is odd then (V (I), 0) has 3 branches, 2 of them smooth and
the third defining an Ak−1 respectively Al−1 singularity. If l and k are odd
then (V (I), 0) has 2 branches, an Ak−1 respectively Al−1 singularity. This
proves the second and third part of the Proposition. For the first part we
have to identify k and l. To do this we blow up of 0 of C3 and consider
the strict transform in the 3 affine charts. We obtain J1 = hy, x2 + z l−2 i,
J2 = hz, x2 + y k−2 i, J3 = hyz, 1 + z l xl−2 + y k xk−2 i. (V (J1 ), 0) is an Al−3
singularity and (V (J2 ), 0) is an Ak−3 singularity.
Algorithm 1 Psingularity(I)
Input: I =< f, g >∈< x, y, z >2 C[[x, y, z]] and 2-jet of I
having normal form (xy, z 2 )
Output: the type of the singularity
1: compute µ =Milnor number of the I;
2: compute τ =Tjurina number of the I;
3: compute B =semigroups of I corresponding to the branches;
4: T = f indlk(I); ∗
5: if µ = τ and µ = T [1] + T [2] + 1 then
6:
if T [1] and T [2] even and B = h1i, h1i, h1i, h1i then
7:
return (PT [1],T [2] );
8:
if T [1] + T [2] odd and B = h2, T [1]i, h1i, h1i or B = h2, T [2]i, h1i, h1i
then
9:
return (PT [1],T [2] );
10:
if T [1] and T [2] odd and B = (2, T [1]), (2, T [2]) then
11:
return (PT [1],T [2] );
12: return (not unimodular);
∗ findlk(I)
is a procedure which computes k and l for the given I = hxy, x2 + y l + z k i.
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Assume the 2-jet of hf, gi has normal form hxy, xzi. According to C.T.C.
Wall’s classification all unimodular curve singularities are in the µ-constant
strata of the versal deformation of the curve singularities given in the Table 3.
Table 3:
Type
F T4,4
F Tk,l
Normal Form
hxy + z 3 , xz + y 3 + λyz 2 i
hxy + z l−1 , xz + y k−1 + yz 2 i
µ
10
l+k+2
τ
10
l+k+1
13
14
16
13
14
16
16 + i
16 + i − 2
h1i, h1i, h1i, h2, 3i
if i odd
h1i, h2, 3i, h2, µ − 13i
if i even
16 + i
14 + i − 2
h1i, h2, 3i, h2, 3i
18
19
6m + 6
6m + 7
6m + 8
6m + 4
18
19
6m + 6
6m + 7
6m + 8
6m + 4
if i even
h1i, h4, 6, τ − 2, τ i
if i odd
h1i, h1i, h3, 5i
h1i, h4, 7, 17i
h1i, h1i, h3, 3m + 1i
h1i, h1i, h2, 2m + 1i
h1i, h1i, h3, 3m + 2i
h1i, h1i, h1i, h2, 2m + ii
6m + 4 + i
5m + 4 + i
k ≥ l ≥ 4, k > 4
F W13
F W14
F W1,0
F W1,i
0
F W1,i
F W18
F W19
F Z6m+6
F Z6m+7
F Z6m+8
F Zm−1,0
F Zm−1,i
hxy + z 3 , xz + y 4 i
hxy + z 3 , xz + zy 3 i
hxy + z 3 , xz + z 2 y 2 + λy 5 i
λ 6= 0, −1/4
hxy + z 3 , xz + z 2 y 2 + y 5+i i
hxy + z 3 , xz + 2z 2 y 2 − y 5
+zy 2 li (z, y)i
hxy + z 3 , xz + zy 4 i
hxy + z 3 , xz + y 6 i
hxy, xz + z 3 + y 3m+1 i
hxy, xz + z 3 + zy 2m+1 i
hxy, xz + z 3 + y 3m+2 i
hxy, xz + z 3 + z 2 y m + λy 3m i
λ 6= 0, −4/27
hxy, xz + z 3 + z 2 y m + y 3m+i i
Semigroup
h1i, h1i, h1i, h1i, h1i
h1i, h1i, h1i, h1i, h1i
if k, l even
h1i, h2, l − 2i, h2, k − 2i
if k, l odd
h1i, h1i, h1i, h2, k − 2i
if k odd, l even
h1i, h4, 5, 11i
h1i, h1i, h3, 4i
h1i, h2, 3i, h2, 3i
h1i, h1i, h1i, h2, 2m + ii
if i odd
h1i, h1i, h1i, h1i, h1i
if i even
Proposition 3.2. The unimodular complete intersection curve singularities
with Milnor number 13, 2 branches and semigroup h1i, h4, 5, 11i are F W13
with Tjurina number 13 defined by the ideal hxy + z 3 , xz + y 4 i and F W13,1
with Tjurina number 12 defined by the ideal hxy + z 3 , xz + y 4 + y 2 z 2 i.
Proof. In the list of C.T.C. Wall F W13 defined by the ideal hxy + z 3 , xz + y 4 i
is the only singularity with µ = 13, 2 branches and semigroup h1i, h4, 5, 11i.
The versal deformation of F W13 is given by hxy + z 3 + ν1 z 2 + ν2 z + ν3 , xz +
4
y + λ1 y 2 z 2 + λ2 yz 2 + λ3 z 2 + λ4 y 2 z + λ5 yz + λ6 z + λ7 y 3 + λ8 y 2 + λ9 i. F W13
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defines a weighted homogenous isolated complete intersection singularity with
weights (w1 , w2 , w3 ) = (11, 4, 5) and degrees (d1 , d2 ) = (15, 16). The versal
µ-constant deformation of F W13 is given by hxy + z 3 , xz + y 4 + λ1 y 2 z 2 i.
Using the coordinate change x → ξ 11 x, y → ξ 4 y, z → ξ 5 z we have Iλ =
hxy + z 3 , xz + y 4 + ξ 2 λ1 y 2 z 2 i. Choose ξ such that ξ 2 λ1 = 1. So we obtain
hxy + z 3 , xz + y 4 + y 2 z 2 i. It has 2 branches and same semigroup as F W13 and
τ = 12.
It can be distinguished from F W13 by the Tjurina number.
Proposition 3.3. The unimodular complete intersection curve singularities
with Milnor number 14, 3 branches two of them are smooth and the third
branch has semigroup h3, 4i are F W14 with Tjurina number 14 defined by the
ideal hxy + z 3 , xz + zy 3 i and F W14,1 with the Tjurina number 13 defined by
the ideal hxy + z 3 , xz + zy 3 + y 5 i.
Proposition 3.4. The unimodular complete intersection curve singularities
with Milnor number 16, 3 branches and semigroup h1i, h2, 3i, h2, 3i are F W1,0
with Tjurina number τ = 16 defined by the ideal hxy + z 3 , xz + z 2 y 2 + λy 5 i
and F W1,0,1 Tjurina number τ = 15 defined by the ideal hxy + z 3 , xz + z 2 y 2 +
λy 5 + y 6 i.
Proof. The proofs of propositions 3.3 and 3.4 are similar to the proof of proposition 3.2.
Proposition 3.5. The unimodular complete intersection curve singularities
with Milnor number 18, 3 branches and semigroup h1i, h1i, h3, 5i are F W18 with
Tjurina number τ = 18 defined by the ideal hxy + z 3 , xz + zy 4 i τ = 18, F W18,1
with Tjurina number τ = 17 defined by the ideal hxy + z 3 , xz + zy 4 + y 7 i and
F W18,2 with Tjurina number τ = 16 defined by the ideal hxy+z 3 , xz+zy 4 +y 6 i.
Proof. In the list of C.T.C. Wall F W18 defined by the ideal hxy + z 3 , xz + zy 4 i
is the only singularity with µ = 18, 3 branches and semigroup h1i, h1i, h3, 5i.
The versal deformation of F W18 is given by hxy + z 3 + ν1 z 2 + ν2 z + ν3 , xz +
4
y z +λ1 y 2 z 2 +λ2 yz 2 +λ3 z 2 +λ4 y 3 z +λ5 y 2 z +λ6 yz +λ7 z +λ8 y 7 +λ9 y 6 +λ10 y 5 +
λ11 y 4 + λ12 y 3 + λ13 y 2 + λ14 y + λ15 i. F W18 defines a weighted homogenous
isolated complete intersection singularity with weights (w1 , w2 , w3 ) = (12, 3, 5)
and degrees (d1 , d2 ) = (15, 17). The versal µ-constant deformation of F W18 is
given by hxy + z 3 , xz + y 4 z + λ8 y 7 + λ9 y 6 i.
If λ9 6= 0 then we have I = hxy + z 3 , xz + y 4 z + uy 6 i where u = λ8 y + λ7 .
Using the coordinate change x → ξ 12 x, y → ξ 3 y, z → ξ 5 z we may assume
I = hxy + z 3 , xz + y 4 z + ξ ūy 6 i. Choose ξ such that ξ ū = 1. So I = hxy +
z 3 , xz + y 4 z + y 6 i. It has Tjurina number τ = 17, two branches and the same
semigroup as F W18 .
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If λ9 = 0 we again apply the same transformation and obtain I = hxy +
z 3 , xz + y 4 z + y 7 i by choosing λ48 = 1. It has Tjurina number τ = 16, also
3 branches and the same semigroup as F W18 . F W18,1 and F W18,2 can be
differentiated from F W18 by the Tjurina numbers.
Proposition 3.6. The unimodular complete intersection curve singularities
with Milnor number 19, 2 branches and semigroup h1i, h4, 7, 17i are F W19 with
Tjurina number τ = 19 defined by the ideal hxy + z 3 , xz + y 6 i, F W19,1 with
Tjurina number τ = 18 defined by the ideal hxy+z 3 , xz+y 6 +y 4 z 2 i and F W19,2
with Tjurina number τ = 17 defined by the ideal hxy + z 3 , xz + y 6 + y 3 z 2 i .
Proof. The proof can be done similarly to the proof of proposition 3.5.
Proposition 3.7. The unimodular complete intersection singularities having
Milnor number of the form µ = 6m + 6 where m is a positive integer with 3
branches , 2 branches are smooth and third branch has semigroup h3, 3m + 1i
are F Z6m+6 defined by the ideal hxy, xz + z 3 + y 3m+1 i with Tjurina number
τ = 6m + 6 and F Z6m+6,i+1 , i=0,1,...,m-1, defined by the ideal hxy, xz + z 3 +
y 3m+1 + y 3m−i zi with Tjurina number τ = µ − i − 1.
Proof. In C.T.C. Wall’s list F Z6m+6 , m ≥ 1 defined by the ideal hxy, xz +
z 3 + y 3m+1 i are the singularities with µ = 6m + 6, 3 branches and semigroup
h1i, h1i, h3, 3m + 1i.
The versal deformation of F Z6m+6 is given by hxy+ν1 z 2 +ν2 z+ν3 , xz+z 3 +
3m
3m
P
P
y 3m+1 +
αi y 3m−i z +
βi y 3m−i i. F Z6m+6 defines a weighted homogenous
i=0
i=0
isolated complete intersection singularity with weights (w1 , w2 , w3 ) = (6m +
2, 3, 3m + 1) and degrees (d1 , d2 ) = (6m + 5, 9m + 3). The versal µ-constant
m−1
P
deformation of F Z6m+6 is given by hxy, xz + z 3 + y 3m+1 +
αi y 3m−i zi.
i=0
Consider φ ∈ AutC (C[[x, y, z]]) defined by φ(x) = ξ 6m+2 x, φ(y) = ξ 3 y
and φ(z) = ξ 3m+1 z. If αm−1 6= 0, then I = hxy, xz + z 3 + y 3m+1 + y 2m+1 z
m−2
m−2
P
P
(
αi y (m−1)−i + αm−1 )i. Let um−1 =
αi y (m−1)−i + αm−1 then I =
i=0
i=0
hxy, xz + z 3 + y 3m+1 +y 2m+1 zum−1 i. By applying the transformation φ we
get I = hxy, xz + z 3 + y 3m+1 + ξy 2m+1 z ūm−1 i. Choose ξ such that ξ ūm−1 = 1
This implies I = hxy, xz + z 3 + y 3m+1 + y 2m+1 zi.
Now we assume αm−1 = 0. This implies I = hxy, xz + z 3 + y 3m+1 + y 2m+2 z
m−3
m−3
P
P
(
αi y (m−2)−i + αm−2 i. Let um−2 =
αi y (m−2)−i + αm−2 then I =
i=0
i=0
hxy, xz + z 3 + y 3m+1 + y 2m+2 zum−2 i. After applying φ we may assume that
I = hxy, xz + z 3 + y 3m+1 + ξ 4 y 2m+2 z ūm−2 i. Choose ξ such that ξ 4 ūm−2 = 1.
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This implies I = hxy, xz + z 3 + y 3m+1 + y 2m+2 zi. If αm−2 = 0 then we assume
αm−3 6= 0.
We may iterate this process and we get m different unimodular singularities
I = hxy, xz +z 3 +y 3m+1 +y 3m−i zi, i = 0, 1, ...., m−1 having Tjurina number
τ = µ − i − 1 and the same semigroup as F Z6m+6 . These singularities can be
distinguished from F Z6m+6 by the Tjurina numbers.
Proposition 3.8. The unimodular complete intersection singularities having
Milnor number of the form µ = 6m + 7 where m is a positive integer with 4
branches , 3 branches are smooth and the fourth branch has semigroup generated by h2, 2m + 1i are F Z6m+7 with Tjurina number τ = 6m + 7 defined
by the ideal hxy, xz + z 3 + zy 2m+1 i and F Z6m+7,i , i = 1, ..., m with Tjurina
number τ = µ − i defined by the ideal hxy, xz + z 3 + zy 2m+1 + y 4m+2−i i .
Proposition 3.9. The unimodular complete intersection singularities having
Milnor number of the form µ = 6m + 8 where m is a positive integer with
3 branches and the semigroup h1i, h1i, h3, 3m + 2i are F Z6m+8 defined by the
ideal hxy, xz + z 3 + y 3m+2 i with Tjurina number τ = 6m + 8 and F Z6m+8,i ,
i = 1, ..., m defined by the ideal hxy, xz + z 3 + y 3m+2 + y 3m+2−i zi with Tjurina
number τ = µ − i.
Proof. The proofs of propositions 3.8 and 3.9 are similar to proof of proposition
3.7.
Summarizing the results of the above propositions we complete the list of
unimodular complete intersection singularities in case of hf, gi having 2-jet
with normal form hxy, xzi.
Proposition 3.10. Let (V (hf, gi), 0) ⊆ (C3 , 0) be the germ of a complete intersection space curve singularity. Assume it is not a hypersurface singularity
and the 2-jet of hf, gi has normal form hxy, xzi. (V (hf, gi), 0) is unimodular
if and only if it is isomorphic to a complete intersection in Table 3 and 4.
Proof. The proof is a direct consequence of C.T.C. Wall’s classification and
Propositions 3.2 - 3.9.
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Algorithm 2 Fsingularity(I)
Input: I =< f, g >∈< x, y, z >2 C[[x, y, z]] and 2-jet of I
having normal form (xy, xz).
Output: the type of the singularity
1: compute µ =Milnor number of the I;
2: compute τ =Tjurina number of the I;
3: compute B =semigroups of I corresponding to the branches;
4: if µ = 10 and B = h1i, h1i, h1i, h1i, h1i then
5:
if µ = τ then
6:
return (F T4,4 );
7: if µ = 13 and B = h1i, h4, 5, 11i then
8:
if µ = τ then
9:
return (F W13 );
10:
if µ − τ = 1 then
11:
return (F W13,1 );
12: if µ = 14 and A = h1i, h1i, h3, 4i then
13:
if µ = τ then
14:
return (F W14 );
15:
if µ − τ = 1 then
16:
return (F W14,1 );
17: if µ = 18 and A = h1i, h1i, h3, 5i then
18:
if µ = τ then
19:
return (F W18 );
20:
else
21:
return (F W18,µ−τ );
22: if µ = 19 and B = h1i, h4, 7, 17i then
23:
if µ = τ then
24:
return (F W19 );
25:
else
26:
return (F W19,µ−τ );
27: if µ = 16 and B = h1i, h2, 3i, h2, 3i then
28:
if µ = τ then
29:
return (F W1,0 );
30:
if µ − τ = 1 then
31:
return (F W1,0,1 );
32: if µ ≡ 0 mod 6 , µ > 11 and B = h1i, h1i, h3, 3(µ − 6)/6 + 1i then
33:
if µ = τ then
34:
return (F Zµ );
35:
else
36:
return (F Zµ,µ−τ );
A CLASSIFIER FOR UNIMODULAR ISOLATED COMPLETE INTERSECTION
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†
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
21:
22:
23:
24:
25:
26:
27:
28:
29:
30:
31:
32:
33:
34:
35:
36:
37:
38:
39:
40:
41:
if µ ≡ 1 mod 6, µ > 12 and B = h1i, h1i, h2, 2(µ − 7)/6 + 1i then
if µ = τ then
return (F Zµ );
else
return (F Zµ,µ−τ );
if µ ≡ 2 mod 6, µ > 13 and B = h1i, h1i, h3, 3(µ − 8)/6 + 2i then
if µ = τ then
return (F Zµ );
else
return (F Zµ,µ−τ );
if µ ≡ 4 mod 6 and µ > 9 then
m = (µ − 4)/6;
if µ = τ then
if m is even and B = h1i, h1i, h1i, h1i then
return (F Zm−1,0 );
if m is odd and B = h1i, h1i, h2, 3(µ − 4)/6i then
return (F Zm−1,0 );
if µ 6= τ then
if µ − τ = 1 then
T = f indlk(I);
if µ = T [1] + T [2] + 2 then
if T [1] and T [2] even and B = h1i, h1i, h1i, h1i, h1i then
return (F TT [1],T [2] );
if T [1] + T [2] odd and B = h1i, h1i, h1i, h2, k − 2i or B =
h1i, h1i, h1i, h2, l − 2i then
return (F TT [1],T [2] );
if T [1] and T [2] odd and B = h1i, h2, l − 2i, h2, k − 2i then
return (F TT [1],T [2] );
if µ is odd and µ > 16 then
if B = h1i, h1i, h1ih2, 3i then
return (F W1,µ−16 );
if B = h1i, h4, 6, µ − 4, µ − 2i then
0
return (F W1,µ−16 );
if µ is even and µ > 16 then
if B = h1i, h2, 3, h2, µ − 13i then
return (F W1,µ−16 );
if B = h1i, h2, 3i, h2, 3i then
0
return (F W1,µ−16 );
if µ is even, µ ≥ 11 and B = h1, h1i, h1i, h1i then
return (F Zµ−τ −1,6τ −5µ−4 );
if µ is odd, µ ≥ 11 and B = h1, h1i, h1i, h1i, h4τ − 3µ − 4, 2i then
return (F Zµ−τ −1,6τ −5µ−4 );
A CLASSIFIER FOR UNIMODULAR ISOLATED COMPLETE INTERSECTION
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Table 4:
Type
Normal Form
µ
τ
Semigroup
F W13,1
hxy + z 3 , xz + y 4 + y 2 z 2 i
13
12
h1i, h4, 5, 11i
F W14,1
hxy + z 3 , xz + zy 3 + y 5 i
14
13
h1i, h1i, h3, 4i
F W1,0,1
hxy + z 3 , xz + z 2 y 2 + λy 5 + y 6 i
λ 6= 0, −1/4
16
15
h1i, h2, 3i, h2, 3i
F W18,1
F W18,2
hxy + z 3 , xz + zy 4 + y 7 i
hxy + z 3 , xz + zy 4 + y 6 i
18
18
17
16
h1i, h1i, h3, 5i
h1i, h1i, h3, 5i
F W19,1
F W19,2
hxy + z 3 , xz + y 6 + y 4 z 2 i
hxy + z 3 , xz + y 6 + y 3 z 2 i
19
19
18
17
h1i, h4, 7, 17i
h1i, h4, 7, 17i
F Z6m+6,i
hxy, xz + z 3 + y 3m+1 + y 3m−i zi
i = 0, 1, ...., m − 1
6m + 6
6m + 5 − i
h1i, h1i, h3, 3m + 1i
F Z6m+7,i
hxy, xz + z 3 + zy 2m+1 + y 4m+2−i i
i = 1, ...., m
6m + 7
6m + 7 − i
h1i, h1i, h2, 2m + 1i
F Z6m+8,i
hxy, xz + z 3 + y 3m+2 + y 3m+2−i zi
6m + 8
6m + 8 − i
h1i, h1i, h3, 3m + 2i
i = 1, ...., m
Proposition 3.11. Assume the 2-jet of hf, gi has normal form hx2 ,y 2 i. According to C.T.C. Wall’s classification the unimodular space curve singularities
are given in the table below
Table 5:
Type
Normal Form
µ
τ
Semigroup
G2n+3 n ≥ 3
(x2 + z 3 , y 2 + z n )
2n + 3
2n + 3
(2, 3), (2, 3)
G2n+6 n ≥ 2
(x2 + z 3 , y 2 + xz n )
2n + 6
2n + 6
(4, 6, 2n + 3)
A CLASSIFIER FOR UNIMODULAR ISOLATED COMPLETE INTERSECTION
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Algorithm 3 Gsingularity(I)
Input: I =< f, g >∈< x, y, z >2 C[[x, y, z]] having 2-jet
of the form (x2 , y 2 ).
Output: the type of the singularity
1: compute µ =Milnor number of I;
2: compute τ =Tjurina number of the I;
3: compute B =semigroups of I corresponding to the branches;
4: if µ = τ then
5:
if µ is even and B = (4, 6, µ − 3) then
6:
return (Gµ );
7:
if µ is odd and B = (2, 3), (2, 3) then
8:
return (Gµ );
9: return (not unimodular);
Assume the 2-jet of hf, gi has normal form hxy, x2 i. According to C.T.C.
Wall’s classification all unimodular curve singularities are in the µ-constant
strata of the versal deformation of the curve singularities given in the table
below
Table 6:
Type
Normal Form
µ
τ
Semigroup
HA11
hxy + z 3 , x2 + z 3 + yz 2 + y 3 i
11
11
h1i, h1i, h1i, h2, 3i
HAr+11
hxy + z 3 , x2 + z 3 + yz 2 + y 3+r i
r≥1
r + 11
r + 10
h1i, h1i, h1i, h2, 3i, µ odd
h1i, h2, 3i, h2, 2 + ri, µ even
HBr+13
hxy + z 3 , x2 + yz 2 + y 4+r i
r≥0
r + 13
r + 12
h3, 4, 5i, h2, 3 + ri
HC13
hxy + z 3 , x2 + z 3 + y 4 i
13
13
h2, 3i, h3, 4i
HC14
hxy + z 3 , x2 + z 3 + zy 3 i
14
14
h1i, h2, 3i, h2, 3i
HC15
hxy + z 3 , x2 + z 3 + y 5 i
15
15
h2, 3i, h3, 5i
HD13
hxy + z 3 , x2 + zy 2 i
13
13
h1i, h4, 5, 7i
HD14
hxy + z 3 , x2 + y 3 i
14
14
h5, 6, 9i
Proposition 3.12. The unimodular complete intersection curve singularities
with Milnor number 13, 2 branches and with semigroups h2, 3i, h3, 4i are HC13
with Tjurina number 13 defined by the ideal hxy + z 3 , x2 + z 3 + y 4 i and HC13,1
with Tjurina number τ = 12 defined by the ideal hxy + z 3 , x2 + z 3 + y 4 + y 3 zi.
A CLASSIFIER FOR UNIMODULAR ISOLATED COMPLETE INTERSECTION
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The singularities having the same Milnor number but being irreducible with
semigroup h5, 6, 9i are HD13 defined by the ideal hxy + z 3 , x2 + zy 2 i with
Tjurina number 13 and HD13,1 with Tjurina number τ = 12 defined by the
ideal hxy + z 3 + yz 2 , x2 + zy 2 + zy 13 i.
Proof. In the list of C.T.C. Wall HC13 defined by the ideal I = hxy + z 3 , x2 +
z 3 + y 4 i is the only singularity with µ = 13, 2 branches and semigroup
h2, 3i, h3, 4i.
We may choose an automorphism ψ ∈ AutC (C[[x, y, z]]) such that ψ(I) =
hxy, x2 + z 3 + y 4 i. The versal deformation is given by hxy + ν1 z 2 + ν2 yz + ν3 z +
ν4 y+ν5 , x2 +z 3 +y 4 +λ1 y 3 z+λ2 y 2 z+λ3 yz+λ4 z+λ5 y 3 +λ6 y 2 +λ7 y+λ8 i. HC13
defines a weighted homogenous isolated complete intersection singularity with
weights hw1 , w2 , w3 i = h6, 3, 4i and degrees hd1 , d2 i = h9, 12i. The versal µconstant deformation of HC13 is given by hxy, x2 + z 3 + y 4 + λ1 y 3 zi. Using the
coordinate change x → ξ 6 x, y → ξ 3 y, z → ξ 4 z, we may assume Iλ1 = hxy +
z 3 , x2 +z 3 +y 4 +ξλ1 y 3 zi. Choose ξ such that ξλ1 = 1. So hxy, x2 +z 3 +y 4 +y 3 zi
has two branches with same semigroup as HC13 and Tjurina number τ = 12.
It can be differentiated from HC13 by the Tjurina number.
In C.T.C. Wall’list HD13 defined by the ideal hxy + z 3 , x2 + zy 2 i is the
only singularity with µ = 13, 2 branches and semigroup h1i, h4, 5, 7i.
The versal deformation of HD13 is given by hxy +z 3 +ν1 yz 2 +ν2 z 2 +ν3 yz +
ν4 z + ν5 y + ν6 , x2 + zy 2 + λ1 z 3 + λ2 yz 2 + λ3 z 2 + λ4 yz + λ5 z + λ6 y + λ7 i. HD13
defines a weighted homogenous isolated complete intersection singularity with
weights hw1 , w2 , w3 i = h7, 5, 4i and degrees hd1 , d2 i = h12, 14i. The versal
µ-constant deformation of HD13 is given by h xy + z 3 + ν1 yz 2 , x2 +y 2 z i.
Using the coordinate change x → ξ 7 x, y → ξ 5 y, z → ξ 4 z, we may assume
Iλ = hxy + z 3 + ξν1 yz 2 , x2 + y 2 zi. Choose ξ such that ξν1 = 1. So hxy +
z 3 + yz 2 , x2 + y 2 zi has 2 branches with same semigroup as HD13 and Tjurina
number τ = 12. It can be differentiated from HD13 by the Tjurina number.
Proposition 3.13. The unimodular complete intersection curve singularities
with Milnor number 14, 3 branches and with semigroup h1i, h2, 3i, h2, 3i are
HC14 with Tjurina number 14 defined by the ideal hxy + z 3 , x2 + z 3 + zy 3 i
and HC14,1 with Tjurina number τ = 13 defined by the ideal hxy + z 3 , x2 +
z 3 + zy 3 + y 5 i. The singularities having same Milnor number but irreducible
with semigroup h5, 6, 9i are HD14 defined by the ideal hxy + z 3 , x2 + y 3 i with
Tjurina number 13 and HD14,1 with Tjurina number τ = 13 defined by the
ideal hxy + z 3 , x2 + y 3 + z 4 i.
Proposition 3.14. The unimodular complete intersection curve singularities
with Milnor number 15, 2 branches and semigroup h2, 3i, h3, 5i are HC15 with
A CLASSIFIER FOR UNIMODULAR ISOLATED COMPLETE INTERSECTION
SPACE CURVE SINGULARITIES
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Tjurina number 15 defined by the ideal hxy + z 3 , x2 + z 3 + y 5 i and HC15,1 with
Tjurina number τ = 14 defined by the ideal hxy + z 3 , x2 + z 3 + y 5 + y 4 zi.
Proof. The proofs of Propositions 3.13 and 3.14 are similar to proof of Proposition 3.12.
Summarizing the results of the propositions above we complete the list of
unimodular complete intersection singularities in case of hf, gi having 2-jet
with normal form hxy, x2 i.
Table 7:
Type
Normal Form
µ
τ
Semigroup
HC13,1
hxy + z 3 , x2 + z 3 + y 4 + y 3 zi
13
12
h2, 3i, h3, 4i
HC14,1
hxy + z 3 , x2 + z 3 + zy 3 + y 5 i
14
13
h1i, h2, 3i, h2, 3i
HC15,1
hxy + z 3 , x2 + z 3 + y 5 + y 4 zi
15
14
h2, 3i, h3, 5i
HD13,1
hxy + z 3 + yz 2 , x2 + zy 2 i
13
12
h1i, h4, 5, 7i
HD14,1
hxy + z 3 , x2 + y 3 + z 4 i
14
13
h5, 6, 9i
Proposition 3.15. Let (V (hf, gi), 0) ⊆ (C3 , 0) be the germ of a complete intersection space curve singularity. Assume it is not a hypersurface singularity
and the 2-jet of hf, gi has normal form hxy, x2 i. (V (hf, gi), 0) is unimodular
if and only if it is isomorphic to a complete intersection in the Table 6 and 7.
Proof. The proof is a direct consequence of C.T.C. Wall’s classification and
Propositions 3.12 - 3.14.
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Algorithm 4 Hsingularity(I)
Input: I =< f, g >∈< x, y, z >2 C[[x, y, z]] 2-jet of I having normal form
(xy, x2 )
Output: the type of the singularity
1: compute µ =Milnor number of the I and τ =Tjurina number of the I;
2: compute B =semigroups of I corresponding to the branches;
3: if µ = 13 then
4:
if µ = τ then
5:
if B = h2, 3i, h3, 4i then
6:
return (HC13 );
7:
if B = h1i, h4, 5, 7i then
8:
return (HD13 );
9:
else
10:
if B = h2, 3i, h3, 4i then
11:
return (HC13,µ−τ );
12:
if B = h1i, h4, 5, 7i then
13:
return (HD13,µ−τ );
14: if µ = 14 then
15:
if µ = τ then
16:
if B = h1i, h2, 3i, h2, 3i then
17:
return (HC14 );
18:
if B = h5, 6, 9i then
19:
return (HD14 );
20:
else
21:
if B = h1i, h2, 3i, h2, 3i then
22:
return (HC14,µ−τ );
23:
if B = h5, 6, 9i then
24:
return (HD14,µ−τ );
25: if µ = 15 and B = h2, 3i, h3, 5i then
26:
if µ = τ then
27:
return (HC15 );
28:
else
29:
return (HC15,µ−τ );
30: if µ = 11 and A = h1i, h1i, h1i, h2, 3i then
31:
if µ = τ then
32:
return (HA11 );
33: if µ 6= τ then
34:
if µ − τ = 1 and µ > 11 then
35:
if µ is even and B = h1i, h2, 3i, h2, µ − 9i then
36:
return (HAµ );
37:
if µ is odd and B = h1i, h1i, h1i, h2, 3i then
38:
return (HAµ );
39:
if µ is even and B = h1i, h1i, h3, 4, 5i then
40:
return (HBµ );
41:
if µ is odd and B = h3, 4, 5i, h2, µ − 10i then
42:
return (HBµ );
43: return not unimodular;
A CLASSIFIER FOR UNIMODULAR ISOLATED COMPLETE INTERSECTION
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Assume the 2-jet of hf, gi has normal form hxy + z 2 , xzi. According to
C.T.C. Wall’s classification all unimodular curve singularities are in the µconstant strata of the versal deformation of the curve singularities given in
the table below
Table 8:
Type
Normal Form
µ
τ
Semigroup
J6m+7
hxy + z 2 , xz + y 3m+3 i
λ 6= 0, −4/27
6m + 7
6m + 7
h1i, h3, 3m + 4, 6m + 5i
J6m+8
hxy + z 2 , xz + zy 2m+2 i
6m + 8
6m + 8
h1i, h1i, h2, 2m + 3i
J6m+9
hxy + z 2 , xz + y 3m+4 i
6m + 9
6m + 9
h1i, h3m + 5, 6m + 7i
Jm+1,0
hxy + z 2 , xz + z2y m + λy 3m+2 i
6m + 5
6m + 5
h1i, h1i, h1i, h1i
Jm+1,i
hxy + z 2 , xz + z2y m + y 3m+2+i i
6m + i + 5
5m + i + 5
h1i, h1i, h2, 2m + i + 2i,
if i is odd
h1i, h1i, h1i, h1i
if i is even
Proposition 3.16. The unimodular complete intersection singularities having Milnor number of the form µ = 6m + 7 where m is a positive integer with 2
branches and semigroup generated by h1i, h3, 3m+4, 6m+5i are J6m+7 defined
by the ideal hxy +z 2 , xz +y 3m+3 i with Tjurina number τ = 6m+7 and J6m+7,i
, i = 0, ..., m − 1 defined by the ideal hxy + z 2 , xz + y 3m+3 + y (3m+1)−i zi with
Tjurina number τ = µ − i.
Proof. In C.T.C. Wall’s list J6m+7 , m ≥ 1 defined by hxy + z 2 , xz + y 3m+3 i
are the singularities with µ = 6m + 7, 2 branches and semigroup h1i, h3, 3m +
4, 6m + 5i.
The versal deformation of J6m+7 is given by hxy+z 2 +ν1 z+ν2 , xz+y 3m+3 +
3m+1
3m+2
P
P
αi y (3m+1)−i z +
βi y (3m+2)−i i. J6m+7 defines a weighted homogenous
i=0
i=0
isolated complete intersection singularity with weights (w1 , w2 , w3 ) = (6m +
5, 3, 3m + 4) and degrees (d1 , d2 ) = (6m + 8, 9m + 9). The versal µ-constant
m−1
P
deformation of J6m+7 is given by hxy + z 2 , xz + y 3m+3 +
αi y (3m+1)−i zi.
i=0
Consider φ ∈ AutC (C[[x, y, z]] defined by φ(x) = ξ 6m+5 x, φ(y) = ξ 3 y
and φ(z) = ξ 3m+4 z. If αm−1 6= 0 then I = hxy + z 2 , xz + y 3m+3 + y 2m+2 z
m−2
P
(
αi y (m−1)−i + αm−1 )i.
i=0
A CLASSIFIER FOR UNIMODULAR ISOLATED COMPLETE INTERSECTION
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Let um−1 =
m−2
P
αi y (m−1)−i + αm−1 . Then I = hxy + z 2 , xz + y 3m+3 +
i=0
y 2m+2 zui. By applying the transformation φ we get I = hxy +z 2 , xz +y 3m+3 +
ξy 2m+2 z ūm−1 i. Choose ξ such that ξ ūm−1 = 1. This implies I = hxy+z 2 , xz+
y 3m+3 + y 2m+2 zi.
m−3
P
αi y (m−2)−i +
If αm−1 = 0 then I = hxy + z 2 , xz + y 3m+3 + y 2m+3 z(
i=0
αm−2 i. Let um−2 =
m−3
P
αi y
(m−2)−i
z + αm−2 then I = hxy, xz + z 3 + y 3m+1 +
i=0
y 2m+3 zum−2 i. After applying φ we may assume that I = hxy + z 2 , xz +
y 3m+3 + ξ 4 y 2m+3 z ūm−2 i. Choose ξ such that ξ 4 ūm−2 = 1. This implies
I = hxy + z 2 , xz + y 3m+3 + y 2m+3 zi.
If αm−2 = 0. We may iterate this process and we get m different unimodular space curve singularities I = hxy + z 2 , xz + y 3m+3 + y (3m+1)−i zi , i =
0, 1, ...., m − 1 having Tjurina number τ = µ − i. These singularities can be
distinguished from J6m+7 by the Tjurina numbers.
Proposition 3.17. The unimodular complete intersection singularities having Milnor number of the form µ = 6m + 8 where m is a positive integer
with 3 branches and semigroup generated by h1i, h1i, h2, 2m + 3iare J6m+8 with
Tjurina number τ = 6m + 8 defined by the ideal hxy + z 2 , xz + zy 2m+2 i and
J6m+8,i , i = 1, ..., m with Tjurina number τ = µ − i defined by the ideal
hxy + z 2 , xz + zy 2m+2 + y 4m+4−i i.
Proposition 3.18. The unimodular complete intersection singularities having Milnor number of the form µ = 6m + 9 where m is a positive integer
with 2 branches and semigroup generated by h1i, h3m + 5, 6m + 7i are J6m+9
with Tjurina number τ = 6m + 9 defined by the ideal hxy + z 2 , xz + y 3m+4 i
and J6m+9,i , i = 1, ..., m with Tjurina number τ = µ − i defined by the ideal
hxy + z 2 , xz + y 3m+4 + y (3m+3)−i i.
Proof. The proofs of Propositions 3.17 and 3.18 can be done similarly to the
proof of Proposition 3.16.
We complete the list of unimodular singularities in this case as
Proposition 3.19. Let (V (hf, gi), 0) ⊆ (C3 , 0) be the germ of a complete intersection space curve singularity. Assume it is not a hypersurface singularity
and the 2-jet of hf, gi has normal form hxy + z 2 , xzi. (V (hf, gi), 0) is unimodular if and only if it is isomorphic to a complete intersection in Table 8 and
9.
Proof. The proof is a direct consequence of C.T.C. Wall’s classification and
Propositions 3.16 - 3.18.
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Table 9:
Type
Normal Form
µ
τ
Semigroup
J6m+7,i
hxy + z 2 , xz + y 3m+3 + y (3m+1)−i zi
6m + 7
6m + 7 − i
h1i, h3, 3m + 4, 6m + 5i
J6m+8,i
hxy + z 2 , xz + zy 2m+2 + y (4m+4)−i i
6m + 8
6m + 8 − i
h1i, h1i, h2, 2m + 3i
J6m+9.i
hxy + z 2 , xz + y 3m+4 + y (3m+3)−i i
6m + 9
6m + 9 − i
h1i, h3m + 5, 6m + 7i
Assume the 2-jet of hf, gi has normal form hxy + z 2 , x2 i. According to
C.T.C. Wall’s classification all unimodular curve singularities are in the µconstant strata of the versal deformation of the curve singularities given in
the table below
Table 10:
Type
Normal Form
µ
τ
Semigroup
K1,0
hxy + z 2 , x2 + z 2 y + λy 4 i
λ 6= 0, 1/4
11
11
h2, 3i, h2, 3i
K1,i
hxy + z 2 , x2 + z 2 y + y 4 + y 4+i i
11 + i
11 + i − 1
h1i, h1i, h2, 3i, if i is odd
h2, 3i, h2, 3 + ii, if i is even
0
hxy + z 2 , x2 + 2z 2 y + y 4 + zyli hz, yii
11 + i
11 + i − 1
K1,i
h2, 3i, h2, 3i, if i is odd
h2, 3i, h2, 8 + ii, if i is even
K13
hxy + z 2 , x2 + zy 3 i
13
13
h1i, h3, 5, 7i
K14
z 2 , x2
14
14
h4, 7, 10i
hxy +
+
y5 i
Proposition 3.20. The unimodular complete intersection curve singularities
with Milnor number 13, 2 branches and semigroup h1i, h3, 5, 7i are K13 with
Tjurina number 13 defined by the ideal hxy + z 2 , x2 + zy 3 i and K13,1 with
Tjurina number τ = 12 defined by the ideal hxy + z 2 , x2 + zy 3 + y 5 i.
Proof. In the list of C.T.C. Wall K13 defined by the ideal hxy + z 2 , x2 + zy 3 i
is the only singularity with µ = 13, 2 branches and semigroup h1i, h3, 5, 7i.
The versal deformation of K13 is given by hxy + z 2 + ν1 z + ν2 , x2 + zy 3 +
λ1 yz 2 + λ2 z 2 + λ3 y 2 z + λ4 yz + λ5 z + λ6 y 5 + λ7 y 4 + λ8 y 3 + λ9 y 2 + λ10 y + λ11 i.
K13 defines a weighted homogenous isolated complete intersection singularity
with weights (w1 , w2 , w3 ) = (7, 3, 5) and degrees (d1 , d2 ) = (10, 14). The
versal µ-constant deformation of K13 is given by hxy + z 2 , x2 + zy 3 + λ6 y 5 i.
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Algorithm 5 Jsingularity(I)
Input: I =< f, g >∈< x, y, z >2 C[[x, y, z]] and 2-jet of I
having normal form (xy + z 2 , xz)
Output: the type of the singularity
1: compute µ =Milnor number of the I;
2: compute τ =Tjurina number of the I;
3: compute B =Semigroup of I corresponding to each branch.;
4: if µ ≡ 1 mod 6 and B = h1i, h3, 3(µ − 7)/6 + 4, 6(µ − 7)/6 + 5i then
5:
if µ = τ then
6:
return (Jµ );
7:
else
8:
return (Jµ,µ−τ );
9: if µ ≡ 2 mod 6 and B = h1i, h1ih2, 2(µ − 8)/6 + 3i then
10:
if µ = τ then
11:
return (Jµ );
12:
else
13:
return (Jµ,µ−τ );
14: if µ ≡ 3 mod 6 and B = h1i, h3, 3(m − 9)/6 + 5, 6(m − 9)/6 + 7i then
15:
if µ = τ then
16:
return (Jµ );
17:
else
18:
return (Jµ,µ−τ );
19: if µ ≡ 5 mod 6 and B = h1i, h1i, h1i, h1i then
20:
if µ = τ then
21:
return (J(µ−5)/6+1,0 );
22: if µ 6= τ then
23:
if µ is even and B = h1i, h1i, h2, 4τ − 3µ − 3i then
24:
return (Jµ−τ +1,6τ −5µ−5 );
25:
if µ is odd then
26:
if B = h1i, h1i, h1i, h1i then
27:
return (Jµ−τ +1,6τ −5µ−5 );
28: return not unimodular;
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Using the coordinate change x → ξ 7 x, y → ξ 3 y, z → ξ 5 z, we may assume
Iλ6 = hxy + z 2 , x2 + zy 3 + ξλ6 y 5 i. Choose ξ such that ξλ6 = 1. So we obtain
hxy + z 2 , x2 + zy 3 + y 5 i. It has 2 branches and same semigroup as K13 but
τ = 12. It can be differentiated from K13 by the Tjurina number.
Proposition 3.21. The unimodular complete intersection curve singularities
with Milnor number 14, irreducible having semigroup h4, 7, 10i are K14 with
Tjurina number 14 defined by the ideal hxy+z 2 , x2 +y 5 i and K14,1 with Tjurina
number τ = 13 defined by the ideal hxy + z 2 , x2 + y 5 + y 2 z 2 i.
We complete the list of unimodular singularities in this case as
Table 11:
K13,1
hxy + z 2 , x2 + zy 3 + y 5 i
13
12
h1i, h3, 5, 7i
K14,1
hxy + z 2 , x2 + y 5 + y 2 z 2 i
14
13
h4, 7, 10i
Proposition 3.22. Let (V (hf, gi), 0) ⊆ (C3 , 0) be the germ of a complete intersection space curve singularity. Assume it is not a hypersurface singularity
and the 2-jet of hf, gi has normal form hxy + z 2 , x2 i. Then (V (hf, gi), 0) is
unimodular if and only if it is isomorphic to a complete intersection in Table
10 and 11.
Proof. The proof is a direct consequence of C.T.C. Wall’s classification and
Propositions 3.20 and 3.21.
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The following Algorithm is the basis for classifying the unimodular complete intersection curve singularities when char(K) = 0.
4
Singular examples
> ring R=0,(x,y,z),ds;
> ideal I=xy+11y2+9yz+z3,x2+22xy+121y2+18xz+198yz+81z2+z3+y4;
> classifyicis1(I);
HC_13:(xy+z3,x2+z3+y4)
> ideal J=x2+xy+2y2+2xz+z2,x2+2xy+xz+2yz+xy12+y12z;
> classifyicis1(J);
J_6*5+8:(xy+z2,xz+zy12)
References
[ADPG1] Afzal, D.; Pfister, G.: A classifier for simple isolated complete interssection singularities. Research preprint series, ASSMS :532 (2013).
[ADPG2] 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).
[DGPS13] 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).
[GM83] Giusti, M.:
Classification des Singularités Isolées Simples
d’Intersections Complètes. Proc. Symp. Pure Math.40 (1983), 457-494.
[GM75] Greuel, G.-M.:
Der Gauß-Manin-Zusamnenhang isolierter
singuläritaten von vollständigen Durchschnitten. Math. Ann. 214(1975),
235-266.
[GP07] Greuel, G.-M.: Pfister, G.: A Singular Introduction to Commutative
Algebra. Second edition, Springer (2007).
[Wal83] Wall, C.T.C.: Classification of unimodal isolated singularities complete intersections. Proc. Symp. Pure Math. 40, Part2, Amer. Math. Soc.
(1983), 625-640.
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Algorithm 6 Ksingulariy(I)
Input: I =< f, g >∈< x, y, z >2 C[[x, y, z]] having 2-jet
of the form (xy + z 2 , x2 )
Output: the type of the singularity
1: compute µ =Milnor number of the I;
2: compute τ =Tjurina number of the I;
3: compute B =semigroups of I corresponding to the branches;
4: if µ = 13 and B = h1i, h3, 5, 7i then
5:
if µ = τ then
6:
return (K13 );
7:
if µ − τ = 1 then
8:
return (K13,1 );
9: if µ = 14 and B = h4, 7, 10i then
10:
if µ = τ then
11:
return (K14 );
12:
if µ − τ = 1 then
13:
return (K14,1 );
14: if µ = 11 and B = h2, 3i, h2, 3i then
15:
if µ = τ then
16:
return (K1,0 );
17: if µ 6= τ then
18:
if µ − τ = 1 and µ > 11 then
19:
if µ is even then
20:
if B = h1i, h1i, h2, 3i then
21:
return (K1,µ−11 );
22:
if B = h2, 3i, h2, 3i then
0
23:
return (K1,µ−11 );
24:
if µ is odd then
25:
if B = h2, 3i, h2, µ − 8i then
26:
return (K1,µ−11 );
27:
if B = h4, 6, µ − 3i then
0
28:
return (K1,µ−11 );
29: return not unimodular;
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Algorithm 7 classifyicis1(I)[Unimodular curve singularities]
Input: I = hf, gi ⊆ hx, y, zi2 C[[x, y, z]] isolated complete intersection curve
singularity.
Output: The type of the singularity (V (I), 0).
1: compute I2 the 2-jet of I;
Ts
2: compute I2 = i=1 Qi the irredundant primary decomposition over C;
3: compute di =Krull dimension of C[x, y, z]/Qi ;
4: compute hi ∈ Q[t] the Hilbert polynomial corresponding to each Qi ;
5: if s = 2 then
6:
if d1 = d2 = 1 then
7:
if h1 = h2 = 2 then
8:
return Psingularity(I); via Algorithm 1
9:
if h1 = 1 and h2 = 4 then
10:
return (Jsingularity); via Algorithm 5
11:
if d1 = 1, d2 = 2 then
12:
if h1 = 1, h2 = 1 + t then
13:
return (Fsingularity(I)); via Algorithm 2
14:
if h1 = 2 and h2 = 1 + t then
15:
return (Hsingulrity(I)); via Algorithm 4
16: if s = 1 then
17:
compute R the radical of I2
18:
if R3 6⊆ I2 then
19:
return (Gsingularity(I)); via Algorithm 3
20:
else
21:
return (Ksingualrtiy(I)); via Algorithm 6
22: return (not unimodular);
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Deeba Afzal,
Department of Mathematics,
University of Lahore, Near Raiwind Road
Lahore 54600, Pakistan.
Email: deebafzal@gmail.com
Gerhard Pfister,
Department of Mathematics,
University of Kaiserslautern,
Erwin-Schrödinger-Str.
67663 Kaiserslautern
Germany.
Email:pfister@mathematik.uni-kl.de
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