Skip to main content

List of Mather singularities

List of the singularities in Mather’s nice dimensions

A complete and repetition-free list of complex algebras corresponding to the singularities in Mather’s nice dimensions, for l≥0. Recall, that the Mather bond is 6l+8 for l=0,1,2,3, and it is 6l+7 if l>3.

The list contains 55 algebras.

For each algebra below there is a nonempty set of l’s for which that algebra appears corresponding to a singularity C^*->C^{*+l} in the nice dimension range.

Notation: d_i = dim_C(m^i/m^{i+1}) for i=1,2,…

\Sigma^I: Thom-Boardman symbol

For a concrete l≥0, a subset of these algebras appear.

    For example, for l=0 these.
    For example, for l=1 these.
    For l≥18, all but the last seven algebras appear.

Names. Some algebras have standard names: A_k, I_{a,b}=I_{ab}, III_{a,b}=III_{ab} (we suppress commas when convenient). For the rest we suggest a name in red below. In these names A, B, C, D, E refer to Thom-Boardman symbols \Sigma^1, \Sigma^2, \Sigma^3, \Sigma^4, \Sigma^5. The first number after the letter refers to \mu. The second number is simply the position in our list. For example C_{52}=C_{5,2}=C(5,2)=C52 is the 2nd algebra listed with \Sigma^3, \mu=5.

[Mather] J. Mather: Stability of C^{\infty} mappings I, II, III, IV, V, VI; Ann Math, Ann Math, Publ IHES, Publ IHES, Adv Math, Springer LNM; 1968-1971.

[Damon] J. Damon: The Classification of Discrete Algebra Types, CUNY preprint, 1973

[MNB] D. Mond, J.-J. Nuno-Ballesteros: Singularities of Mappings: The Local Behaviour of Smooth and Complex Analytic Mappings; Springer, 2020.

[P] B. Poonen: Isomorphism types of commutative algebras of finite rank over an algebraically closed field; link 2007 (corr. 2020).

\mu=0

    \Sigma^0
    l=0,1,2,…

\mu=1

    \Sigma^1
    l=0,1,2,…
    d=1

\mu=2

    \Sigma^{11}
    l=0,1,2,…
    d=1,1
    \Sigma^{20}
    l=1,2,…
    d=2

\mu=3

    \Sigma^{111}
    l=0,1,2,…
    d=1,1,1
    \Sigma^{20}
    l=0,1,2,…
    d=2,1
    l=1,2,3,…
    d=2,1
    \Sigma^3
    l=3,4,5,…
    d=3
    Name: C(3)

\mu=4

    \Sigma^{1111}
    l=0,1,2,…
    d=1,1,1,1
    \Sigma^{20}
    l=0,1,2,…
    d=2,1,1
    l=1,2,3,…
    d=2,1,1
    l=1,2,3,…
    d=2,2
    \Sigma^{21}
    l=1,2,3,…
    d=2,2
    Name: B_4
    \Sigma^{3}
    l=2,3,4,…
    d=3,1
    D_{3,0} in [MNB] Name: C_{4,1}
    l=2,3,4,…
    d=3,1
    D_{2,0} in [MNB], (x^2-y^2,z^2,xy,yz,zx) — but this representative hides a U(1) symmetry
    Name: C(4,2)
    l=3,4,5…
    d=3,1
    Name: C_{4,3}
    \Sigma^{4}
    l=6,7,8,…
    d=4
    Name: D_{4}

\mu=5

    \Sigma^{11111}
    l=0,1,2,…
    d=1,1,1,1,1
    \Sigma^{20}
    l=0,1,2,…
    d=2,1,1,1
    l=0,1,2,…
    d=2,2,1
    l=1,2,3,…
    d=2,1,1,1
    l=1,2,3,…
    d=2,2,1
    \Sigma^{21}
    l=0,1,2,…
    d=2,2,1
    Name: B_{5,1}
    l=1,2,3,…
    d=2,2,1
    Name: B_{5,2}
    l=1,2,3,…
    d=2,2,1
    Name: B_{5,3}
    \Sigma^{22}
    l=2,3,4,…
    d=2,3
    Name: B_{5,4}
    \Sigma^3
    l=1,2,…
    d=3,2
    [MNB] Table 7.8, line 1
    a.k.a. (x^2,y^2,z^2-xy,xz+yz) in [MNB] Table 7.8 line 2
    Name: C_{5,1}
    l=1,2,3,…
    d=3,2
    [MNB] Table 7.8, line 3
    a.k.a. (x^2,y^2-z^2,xy+xz,yz) in [MNB] Table 7.8 line 4
    a.k.a. (x^2,y^2-z^2,xy,yz) in [P] Table 1
    Name: C_{5,2}
    l=2,3,4,…
    d=3,1,1
    a.k.a. (y^2+x^3,z^2+x^3,xy,xz,yz) in [MNB] Table 7.7, line 4 — this presentation does not show the U(1)^2 symmetry, only U(1)
    a.k.a. (x^2,xy+z^3,y^2,xz,yz,z^4) in [P] Table 1 (in this presentation the last generator is redundant)
    Name: C_{5,3}
    l=1,2,3,4,…
    d=3,2
    [MNB] Table 7.8, line 10
    a.k.a. (xy,z^2,xz-yz,x^2+y^2-xz) in [P] Table 1
    Name: C_{5,4}
    l=2,3,4,…
    d=3,2
    [MNB] Table 7.8 line 5
    a.k.a. (x^2-y^2,xy,xz,yz,z^3) in [MNB] Table 7.8 line 7
    Name: C_{5,5}
    l=2,3,4,…
    d=3,1,1
    [MNB] Table 7.7, line 5
    a.k.a. (x^2,xy,y^2+z^3,xz,yz,z^4) in [P] Table 1 (in this presentation the last generator is redundant)
    Name: C_{5,6}
    l=2,3,4,…
    d=3,2
    [MNB] Table 7.8, line 11
    Name: C_{5,7}
    l=3,4,5,… (for l=1,2 this exists, but is not in the nice dimension range)
    d=3,2
    [P] Table 1
    Name: C_{5,8}
    l=3,4,5,…
    d=3,1,1
    [P] Table 1
    Name: C_{5,9}
    l=3,4,5,…
    d=3,2
    [P] Table 1
    Name: C_{5,10}
    l=5,6,7,… (for l=3,4 this exists, but is not in the nice dimension range)
    d=3,2
    [P] Table 1
    Name: C_{5,11}
    \Sigma^4
    l=5,6,7,…
    d=4,1
    Name: D_{5,1}
    l=5,6,7,…
    d=4,1
    Name: D_{5,2}
    l=7,8,9,… (for l=5,6 this exists, but is not in the nice dimension range)
    d=4,1
    Name: D_{5,3}
    l=10,11,12,… (for l=6,7,8,9 this exists, but is not in the nice dimension range)
    d=4,1
    Name: D_{5,4}
    \Sigma^5
    l=18,19,20,… (for l=10,11,12,13,14,15,16,17 this exists, but is not in the nice dimension range)
    d=5
    Name: E_{5}

\mu=6

    \Sigma^{111111}
    l=0,1,2,…
    d=1,1,1,1,1,1
    \Sigma^{20}
    l=0,1,2,3,…
    d=2,1,1,1,1
    l=0,1,2,3,…
    d=2,2,1,1
    l=1,2,3 only. (For larger l this is not in the nice dimension range.)
    d=2,1,1,1,1
    l=1,2,3 only. (For larger l this is not in the nice dimension range.)
    d=2,2,1,1
    l=1,2,3 only. (For larger l this is not in the nice dimension range.)
    d=2,2,2
    \Sigma^{21}
    l=0,1,2,3 only. (For larger l this is not in the nice dimension range.)
    d=2,2,1,1
    Name: B_{6}
    \Sigma^3
    l=1,2,3 only. (For larger l this is not in the nice dimension range.)
    d=3,2,1
    another presentation: (x^2+y^2+z^3,xy,xz,yz)
    Name: C(6)

\mu=7

    \Sigma^{1111111}
    l=0,1 only. (For larger l this is not in the nice dimension range.)
    d=1,1,1,1,1,1,1
    \Sigma^{20}
    l=0 only. (For larger l this is not in the nice dimension range.)
    d=2,1,1,1,1,1
    l=0 only. (For larger l this is not in the nice dimension range.)
    d=2,2,1,1,1
    l=0 only. (For larger l this is not in the nice dimension range.)
    d=2,2,2,1

\mu=8

    \Sigma^{11111111}
    l=0 only. (For larger l this is not in the nice dimension range.)
    d=1,1,1,1,1,1,1,1