7 dimensional algebras

List of 7 dimensional associative, commutative, local algebras over C

General references: (with some corrections)

[Ga] F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Chelsea Publishing Co., New York, 1959, Translated by K. A. Hirsch

[Ma] 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.

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

[AbEmIa] N. Abdallah, J. Emsalem, A. Iarrobino: Nets of Conics and associated Artinian algebras of length 7, (versions from 1977 to 2022)

[Yu] Alexandria Yu: Towards the classification of unital 7-dimensional commutative algebras, 2014

[Je] J. Jelisiejew: personal communication

[Th] Robert C. Thompson: Pencils of complex and real symmetric and skew matrices, Linear Algebra and its Applications, 147, 1991, 323-371.

[Be] A. Becker: personal communication

There are 2 one-parameter families **, and 77 (or 78 or 79 discrete) types. (The counting depends on whether we consider C^1 or P^1 families, for Nets of Conics and for Pencils of Quadrics, see below.)

This page is mostly complete, except the POQ part.

$\Sigma^1$

$d=(1,1,1,1,1,1)$

(x^7)
l0=0

$\Sigma^2$

$d=(2,1,1,1,1)$

(x*y,x^2+y^5)
l0=0

(x^2,x*y,y^6)
l0=1

$d=(2,2,1,1)$

(x*y,x^3+y^4)
l0=0

(x*y,x^3,y^5)
l0=1

(x^2-y^4,x*y^2,y^5)
l0=1

(x^2,x*y^2,y^5)
l0=1

$d=(2,2,2)$

l0=1

(x^3+y^3,x*y^2)
l0=0

(x^2-y^3,x*y^3,y^4)
l0=1

(x^2,x*y^3,y^4)
l0=1

$d=(2,3,1)$

(x^3,x^2*y-x*y^2,y^3)
l0=1

(x^3,x^2*y,y^3)
l0=1

(x^4,x^2*y,x*y^2,y^3)
l0=2

$\Sigma^3$

$d=(3,1,1,1)$

(x^4-z^2,y^2-z^2,x*y,x*z,y*z)
l0=2
[Dam] p36, Sigma_{r,(1)} for r=3, l=4, k=2

(x^4-z^2,y^2,x*y,x*z,y*z)
l0=2

(x*y,x*z,y*z,y^2,z^2,x^5)
l0=3

$d=(3,2,1)$

(x*y,x*z,y^2-x^3,z^2-x^3)
l0=1
[Yu] Thm. 5.4 (17) (with redundant generator x^4)

(x*y,x*z,y^2,z^2-x^3)
l0=1
[Yu] Thm. 5.4 (16) (with redundant generator x^4)

(x*y,y^2-x*z,y*z,z^2-x^3)
l0=1
[Yu] Thm. 5.4 (3) (with redundant generator x^4)
[MNB] p248, T7.9 line 1

(x*y,x*z,y*z,y^3-z^3,x^2-z^3)
l0=2
[Yu] Thm. 5.4 (13)

(x*y,x*z,y^2,z^2,x^4)
l0=2
[Yu] Thm. 5.4 (15)

(x^2,x*y,x*z,y*z,y^3-z^3)
l0=2
[Yu] Thm. 5.4 (14)

(x*y,x*z,y*z,y^3,x^2-z^3)
l0=2
[Yu] Thm. 5.4 (10)

(x*y,x*z,y^2,z^3,x^2-y*z^2)
l0=2
[Yu] Thm. 5.4 (8)

(x*y,y^2-x*z,z^2,y*z-x^3,x^4)
l0=2
[Yu] Thm. 5.4 (2)

(x^2,x*z,y*z,y^3,x*y-z^3)
l0=2
[Yu] Thm. 5.4 (11)

(x^2,x*y,x*z,y^2,z^3)
l0=2
[Yu] Thm. 5.4 (9)

(x*y,y^2-x*z,z^2,y*z,x^4)
l0=2
[Yu] Thm. 5.4 (1)

(z^2,y^2,x*y-x*z,x^2*z,x^3-y*z)
l0=2
[JJ]

(x*y,x*z,y*z,x^2,y^3,z^4)
l0=3
[Yu] Thm. 5.4 (12)

(y^2,x*z,y*z^2,x*y,x^2-z^3,z^4)
l0=3
[Yu] Thm. 5.4 (6)

(x^2,y^2,x*z,y*z^2,x*y-z^3,z^4)
l0=3
[Yu] Thm. 5.4 (4)

(x^2,x*y,x*z,y^2-z^3,y*z^2,z^4)
l0=3
[Yu] Thm. 5.4 (7)

(x^2,y^2,x*z,y*z^2,x*y,z^4)
l0=3
[Yu] Thm. 5.4 (5)

$d=(3,3)$

Non-degenerate nets of conics (NOC). Their hierarchy.

* This list includes a 1-parameter continuous family, NOC-A(c,g), as well as 12 discrete ones.
(We could take one from the continuous P^1-family to be a discrete one, calling it NOC-B, thus creating a C^1-family. That way there would be 13 discrete ones.)

\mu \in P^1
l0=2 for mu=infty, l0=1 otherwise
Each such algebra is of codimension 6l+10, the pencil together is of codimension 6l+9
The equivariant properties of NOC( $\infty$ ) are slightly different from the others [DFR].

l0=1
(x*y,z^2,y^2-2*x*y,x^3)

l0=2
(x*z,y*z,z^2-2*x*y,x^3,y^3)

l0=1
(x^2,y^2,z^2-2*x*y,z^3)

l0=3
(x*y,x*z,y*z,x^3,y^3,z^3)

l0=1
(x^2,y^2,z^2,x*y*z)

l0=2
(x*z,z^2,x^2-y^2,x^3,x^2*y)

l0=2
(x^2-y^2,x*y,y*z,z^3,x*z^2)

l0=3
(z^2,x*y,x*z,x^3,y^3,y^2*z)

l0=2
(x^2,y^2,y*z,z^3,x*z^2)

l0=3
(y^2-2*x*y,y*z,z^2,x^3,x^2*y,x*y^2)

l0=4
(x*z,y*z,z^2,x^3,x^2*y,x*y^2,y^3)

l0=3
(x^2,x*y,y^2,x*z^2,y*z^2,z^3)

$\Sigma^4$

$d=(4,1,1)$

(x^3-y^2,x^3-z^2,x^3-u^2,x*y,x*z,x*u,y*z,y*u,z*u)
l0=5

(x^3-y^2,x^3-z^2,u^2,x*y,x*z,x*u,y*z,y*u,z*u)
l0=5

(x^3-y^2,z^2,u^2,x*y,x*z,x*u,y*z,y*u,z*u)
l0=5

(x*y,x*z,x*u,y*z,y*u,z*u,y^2,z^2,x^4)
l0=6

$d=(4,2)$

Pencils of quadrics (POQ), both regular and irregular POQs.

* This list includes one 1-parameter family POQ-0*, as well as, 21 discrete ones.
(We could add one of the 21 discrete ones, namely POQ-1, to the C^1-family to create a P^1-family, that way there would be only 20 discrete ones.)

1. Regular pencils of quadrics.

(x*y,x*z,x*u,y*z,y*u,z*u,z^2-x^2-y^2,u^2-x^2-c*y^2)

l0 = 4

Each such is of codimension 6l+9, the family is hence of codimension 6l+8

Segre symbol = [(1),(1),(1),(1)]

(x^2+y^2+z^2, y^2+cz^2+u^2)^c

Fundamental class = 12*(V[1] – U[1])

(x^2,x*z,x*u,y*z,y*u,z*u,z^2-xy-y^2,u^2-xy+y^2)

l0 = 4

Segre symbol = [(2),(1),(1)]

(y^2+z^2+u^2, xy+u^2)^c

Fundamental class = ?*(U[1] – V[1])

(x*y,x*z,x*u,y^2,y*z,y*u,z*u,z^2-u^2)

l0 = 4

Segre symbol = [(1,1),(1),(1)]

(z^2+u^2, xy+u^2)^c

Fundamental class = 6*U[1]^2 – 15*U[1]*V[1] + 10*V[1]^2 + 4*U[2] – 10*V[2]

(x^2,x*y,x*u,y*u,z^2,z*u,y^2,y*z-u^2)

l0 = 4

Segre symbol = [(3),(1)]

(yz+u^2, xz+y^2)^c

Fundamental class = 24*(U[1] – V[1])^2

(x^2,x*z,x*u,y^2,y*u,z^2,u^2,y*z-z*u)

l0 = 4

Segre symbol = [(2),(2)]

(y^2+zu, xy+u^2)^c

Fundamental class = 16*U[1]^2 – 32*U[1]*V[1] + 12*V[1]^2 + 16*V[2]

(x^2,x*z,x*u,y*z,y*u,z*u,z^2,y^2-u^2)

l0 = 4

Segre symbol = [(2,1),1]

(y^2+u^2, xy+z^2)^c

Fundamental class = -2*(U[1] – V[1])*(6*U[1]^2 – 15*U[1]*V[1] + 10*V[1]^2 + 4*U[2] – 10*V[2])

(x^2,x*y,x*z,y^2,z*u,u^2,x*u,z^2)

l0 = 4

Segre symbol = [(4)]

(yu+z^2, xu+yz)^c

Fundamental class = -8*(U[1] – V[1])*(4*U[1]^2 – 8*U[1]*V[1] + 3*V[1]^2 + 4*V[2])

(x^2,x*z,x*u,y*z,y*u,z^2,u^2,y^2)

l0 = 4

Segre symbol = [(2),(1,1)]

(y^2+zu, xy)^c

Fundamental class = -24*U[1]^3 + 72*U[1]^2*V[1] – 70*U[1]*V[1]^2 + 20*V[1]^3 – 20*U[1]*V[2] + 8*U[2]*V[1] + 20*V[1]*V[2] – 16*U[3]

(x^2,x*y,x*u,y*u,z^2,z*u,y^2,u^2)

l0 = 4

Segre symbol = [(3,1)]

(xz+y^2+u^2, yz)^c)

Fundamental class = 2*(U[1] – V[1])*(12*U[1]^3 – 36*U[1]^2*V[1] + 35*U[1]*V[1]^2 – 10*V[1]^3 + 10*U[1]*V[2] – 4*U[2]*V[1] – 10*V[1]*V[2] + 8*U[3])

(x^2,x*z,x*u,y^2,y*z,y*u,z^2,u^2)

l0 = 4

Segre symbol = [(1,1),(1,1)]

(xy, zu)^c

Fundamental class = 12*U[1]^4 – 42*U[1]^3*V[1] + 54*U[1]^2*V[1]^2 – 30*U[1]*V[1]^3 + 6*V[1]^4 – 8*U[1]^2*U[2] + 27*U[1]^2*V[2] + 6*U[1]*U[2]*V[1] – 45*U[1]*V[1]*V[2] + 6*U[2]*V[1]^2 + 12*V[1]^2*V[2] + 20*U[1]*U[3] – 12*U[2]*V[2] – 24*U[3]*V[1] + 21*V[2]^2 + 16*U[4]

(x^2,x*z,x*u,y^2,y*z,y*u,z*u,u^2-z^2)

l0 = 4

Segre symbol =[(1,1,1),(1)]

(u^2, xy+z^2)^c

Fundamental class = -8*(U[1] – V[1])*(U[1]^2*V[1]^2 – 2*U[1]*V[1]^3 + V[1]^4 – U[1]^2*V[2] – 2*U[1]*U[2]*V[1] + 5*U[1]*V[1]*V[2] + 2*U[2]*V[1]^2 – 4*V[1]^2*V[2] + U[1]*U[3] + U[2]^2 – 4*U[2]*V[2] + 3*V[2]^2 – 4*U[4])

(x^2,x*z,x*u,y^2,y*z,y*u,z*u,u^2-z^2)

l0 = 4

Segre symbol = [(2,2)]

(y^2+u^2, xy+zu)^c

Fundamental class = 2*(U[1] – V[1])*(12*U[1]^3*V[1] – 30*U[1]^2*V[1]^2 + 24*U[1]*V[1]^3 – 6*V[1]^4 – 16*U[1]^2*U[2] + 12*U[1]^2*V[2] + 24*U[1]*U[2]*V[1] – 15*U[1]*V[1]*V[2] – 6*U[2]*V[1]^2 + 16*U[1]*U[3] – 12*U[2]*V[2] – 12*U[3]*V[1] + 15*V[2]^2 – 16*U[4])

(x^2,x*z,x*u,y*z,y*u,z^2,u^2,y^2)

l0 = 4

Segre symbol = [(2,1,1)]

(y^2, xy+zu)^c

Fundamental class = 16*U[1]^4*V[1]^2 – 56*U[1]^3*V[1]^3 + 72*U[1]^2*V[1]^4 – 40*U[1]*V[1]^5 + 8*V[1]^6 – 16*U[1]^4*V[2] – 32*U[1]^3*U[2]*V[1] + 88*U[1]^3*V[1]*V[2] + 80*U[1]^2*U[2]*V[1]^2 – 144*U[1]^2*V[1]^2*V[2] – 64*U[1]*U[2]*V[1]^3 + 88*U[1]*V[1]^3*V[2] + 16*U[2]*V[1]^4 – 16*V[1]^4*V[2] + 16*U[1]^3*U[3] + 16*U[1]^2*U[2]^2 – 32*U[1]^2*U[2]*V[2] – 24*U[1]^2*U[3]*V[1] – 24*U[1]*U[2]^2*V[1] + 32*U[1]*U[2]*V[1]*V[2] + 8*U[1]*U[3]*V[1]^2 + 24*U[1]*V[1]*V[2]^2 + 8*U[2]^2*V[1]^2 – 24*V[1]^2*V[2]^2 – 48*U[1]^2*U[4] – 16*U[1]*U[2]*U[3] + 16*U[1]*U[3]*V[2] + 96*U[1]*U[4]*V[1] + 16*U[2]^2*V[2] – 32*U[2]*V[2]^2 – 32*U[4]*V[1]^2 + 16*V[2]^3 + 16*U[3]^2 – 64*U[4]*V[2]

2. Irregular pencils of quadrics.

(x^2,x*y,x*u,y^2,y*u,z^2,z*u,u^2)

l0 = 4

(xz, yz+u^2)^c

Fundamental class = 4*(U[1] – V[1])*(6*U[1]^3*V[1] – 11*U[1]^2*V[1]^2 + 4*U[1]*V[1]^3 – 8*U[1]^2*U[2] – 7*U[1]^2*V[2] + 10*U[1]*U[2]*V[1] + 20*U[1]*V[1]*V[2] – 8*V[1]^2*V[2] + 12*U[1]*U[3] – 8*U[2]*V[2] – 12*U[3]*V[1] – 4*V[2]^2)

[Th][Be]

(x*y,x*z,x*u,y*z,y*u,z^2,u^2,x^2+y^2-z^2)

l0 = 4

(y^2+z^2, x^2+y^2)^c

Fundamental class = -(U[1] – V[1])*(4*U[1]^4 – 10*U[1]^3*V[1] + 8*U[1]^2*V[1]^2 – 2*U[1]*V[1]^3 – 8*U[1]^2*U[2] + 13*U[1]^2*V[2] + 10*U[1]*U[2]*V[1] – 20*U[1]*V[1]*V[2] + 4*V[1]^2*V[2] + 12*U[1]*U[3] – 8*U[2]*V[2] – 12*U[3]*V[1] + 12*V[2]^2)

[Po][Th][Be]

(x^2,x*z,x*u,y*z,y*u,z*u,u^2,y^2 – z^2)
l0 = 4
(xy, y^2+z^2)^c
Fundamental class = -24*U[1]^5*V[1] + 84*U[1]^4*V[1]^2 – 108*U[1]^3*V[1]^3 + 60*U[1]^2*V[1]^4 – 12*U[1]*V[1]^5 + 32*U[1]^4*U[2] – 64*U[1]^3*U[2]*V[1] – 78*U[1]^3*V[1]*V[2] + 36*U[1]^2*U[2]*V[1]^2 + 174*U[1]^2*V[1]^2*V[2] – 4*U[1]*U[2]*V[1]^3 – 120*U[1]*V[1]^3*V[2] + 24*V[1]^4*V[2] – 32*U[1]^3*U[3] – 32*U[1]^2*U[2]^2 + 104*U[1]^2*U[2]*V[2] + 40*U[1]^2*U[3]*V[1] – 24*U[1]^2*V[2]^2 + 32*U[1]*U[2]^2*V[1] – 120*U[1]*U[2]*V[1]*V[2] + 16*U[2]*V[1]^2*V[2] – 8*U[3]*V[1]^3 + 24*V[1]^2*V[2]^2 + 32*U[1]^2*U[4] + 64*U[1]*U[2]*U[3] – 80*U[1]*U[3]*V[2] – 64*U[1]*U[4]*V[1] – 32*U[2]^2*V[2] – 32*U[2]*U[3]*V[1] + 64*U[2]*V[2]^2 + 48*U[3]*V[1]*V[2] – 32*V[2]^3 – 32*U[3]^2 + 128*U[4]*V[2]

(x^2,x*y,x*u,y*u,z^2,z*u,u^2,y^2)

l0 = 4

(yz, xz-y^2)^c

Fundamental class = -3*(U[1] – V[1])*(8*U[1]^4*V[1]^2 – 20*U[1]^3*V[1]^3 + 16*U[1]^2*V[1]^4 – 4*U[1]*V[1]^5 + 4*U[1]^4*V[2] – 24*U[1]^3*U[2]*V[1] – 10*U[1]^3*V[1]*V[2] + 44*U[1]^2*U[2]*V[1]^2 + 22*U[1]^2*V[1]^2*V[2] – 20*U[1]*U[2]*V[1]^3 – 24*U[1]*V[1]^3*V[2] + 8*V[1]^4*V[2] + 16*U[1]^2*U[2]^2 – 8*U[1]^2*U[2]*V[2] + 24*U[1]^2*U[3]*V[1] + U[1]^2*V[2]^2 – 16*U[1]*U[2]^2*V[1] – 20*U[1]*U[2]*V[1]*V[2] – 36*U[1]*U[3]*V[1]^2 + 10*U[1]*V[1]*V[2]^2 + 24*U[2]*V[1]^2*V[2] + 16*U[3]*V[1]^3 – 8*V[1]^2*V[2]^2 – 32*U[1]*U[2]*U[3] + 24*U[1]*U[3]*V[2] + 16*U[2]^2*V[2] + 16*U[2]*U[3]*V[1] – 16*U[2]*V[2]^2 – 24*U[3]*V[1]*V[2] + 4*V[2]^3 + 16*U[3]^2)

[Po] [Th] [Be]

(x^2,x*z,x*u,y^2,y*z,y*u,z*u,u^2,z^3)

l0 = 5

(xy, z^2)^c

Fundamental class = -8*(U[1] – V[1])*(2*U[1]^4*V[1]^2 – 5*U[1]^3*V[1]^3 + 4*U[1]^2*V[1]^4 – U[1]*V[1]^5 – 2*U[1]^4*V[2] – 4*U[1]^3*U[2]*V[1] + 5*U[1]^3*V[1]*V[2] + 6*U[1]^2*U[2]*V[1]^2 + U[1]^2*V[1]^2*V[2] – 2*U[1]*U[2]*V[1]^3 – 6*U[1]*V[1]^3*V[2] + 2*V[1]^4*V[2] + 2*U[1]^3*U[3] + 2*U[1]^2*U[2]^2 + 4*U[1]^2*U[2]*V[2] – U[1]^2*U[3]*V[1] – 8*U[1]^2*V[2]^2 – U[1]*U[2]^2*V[1] – 12*U[1]*U[2]*V[1]*V[2] + 13*U[1]*V[1]*V[2]^2 + 4*U[2]*V[1]^2*V[2] – 2*V[1]^2*V[2]^2 – 2*U[1]^2*U[4] – 6*U[1]*U[2]*U[3] + 2*U[1]*U[3]*V[2] + 4*U[1]*U[4]*V[1] + 2*U[2]^2*V[2] + 4*U[2]*V[2]^2 – 6*V[2]^3 + 6*U[3]^2 – 8*U[4]*V[2])

[Po][Th][Be]

(x^2,x*z,x*u,y*z,y*u,z*u,u^2,z^2-y^2)

l0 = 4

(xy+z^2, y^2)^c

Fundamental class = -16*U[1]^5*V[1]^3 + 56*U[1]^4*V[1]^4 – 72*U[1]^3*V[1]^5 + 40*U[1]^2*V[1]^6 – 8*U[1]*V[1]^7 + 64*U[1]^4*U[2]*V[1]^2 – 24*U[1]^4*V[1]^2*V[2] – 176*U[1]^3*U[2]*V[1]^3 + 32*U[1]^3*V[1]^3*V[2] + 160*U[1]^2*U[2]*V[1]^4 + 24*U[1]^2*V[1]^4*V[2] – 48*U[1]*U[2]*V[1]^5 – 48*U[1]*V[1]^5*V[2] + 16*V[1]^6*V[2] – 16*U[1]^4*U[3]*V[1] – 24*U[1]^4*V[2]^2 – 80*U[1]^3*U[2]^2*V[1] + 96*U[1]^3*U[2]*V[1]*V[2] – 8*U[1]^3*U[3]*V[1]^2 + 72*U[1]^3*V[1]*V[2]^2 + 152*U[1]^2*U[2]^2*V[1]^2 – 96*U[1]^2*U[2]*V[1]^2*V[2] + 88*U[1]^2*U[3]*V[1]^3 – 144*U[1]^2*V[1]^2*V[2]^2 – 72*U[1]*U[2]^2*V[1]^3 – 64*U[1]*U[2]*V[1]^3*V[2] – 96*U[1]*U[3]*V[1]^4 + 144*U[1]*V[1]^3*V[2]^2 + 64*U[2]*V[1]^4*V[2] + 32*U[3]*V[1]^5 – 48*V[1]^4*V[2]^2 + 16*U[1]^4*U[4] + 16*U[1]^3*U[2]*U[3] + 24*U[1]^3*U[3]*V[2] + 8*U[1]^3*U[4]*V[1] + 32*U[1]^2*U[2]^3 – 72*U[1]^2*U[2]^2*V[2] + 88*U[1]^2*U[2]*U[3]*V[1] + 32*U[1]^2*U[2]*V[2]^2 – 208*U[1]^2*U[3]*V[1]*V[2] – 64*U[1]^2*U[4]*V[1]^2 – 8*U[1]^2*V[2]^3 – 32*U[1]*U[2]^3*V[1] – 16*U[1]*U[2]^2*V[1]*V[2] – 160*U[1]*U[2]*U[3]*V[1]^2 + 112*U[1]*U[2]*V[1]*V[2]^2 + 304*U[1]*U[3]*V[1]^2*V[2] + 32*U[1]*U[4]*V[1]^3 – 32*U[1]*V[1]*V[2]^3
+ 80*U[2]^2*V[1]^2*V[2] + 64*U[2]*U[3]*V[1]^3 – 128*U[2]*V[1]^2*V[2]^2 – 128*U[3]*V[1]^3*V[2] + 32*V[1]^2*V[2]^3 – 96*U[1]^2*U[2]*U[4] – 16*U[1]^2*U[3]^2 + 16*U[1]^2*U[4]*V[2] – 64*U[1]*U[2]^2*U[3] + 176*U[1]*U[2]*U[3]*V[2] + 128*U[1]*U[2]*U[4]*V[1] – 24*U[1]*U[3]^2*V[1] – 112*U[1]*U[3]*V[2]^2 + 64*U[1]*U[4]*V[1]*V[2] + 32*U[2]^3*V[2] + 32*U[2]^2*U[3]*V[1] – 80*U[2]^2*V[2]^2 – 128*U[2]*U[3]*V[1]*V[2] + 64*U[2]*V[2]^3 + 32*U[3]^2*V[1]^2 + 96*U[3]*V[1]*V[2]^2 – 64*U[4]*V[1]^2*V[2] – 16*V[2]^4 + 128*U[1]*U[3]*U[4] + 32*U[2]*U[3]^2 – 128*U[2]*U[4]*V[2] – 16*U[3]^2*V[2] – 128*U[3]*U[4]*V[1] + 64*U[4]*V[2]^2

[Po][Th][Be]

(x^2,x*u,y^2,y*z,y*u,z^2,z*u,u^2)

l0 = 4

(xy ,xz)^c

Fundamental class = -24*U[1]^5*V[1]^2*V[2] + 16*U[1]^4*U[2]*V[1]^3 + 60*U[1]^4*V[1]^3*V[2] – 40*U[1]^3*U[2]*V[1]^4 – 48*U[1]^3*V[1]^4*V[2] + 32*U[1]^2*U[2]*V[1]^5 + 12*U[1]^2*V[1]^5*V[2] – 8*U[1]*U[2]*V[1]^6 + 12*U[1]^5*V[2]^2 + 48*U[1]^4*U[2]*V[1]*V[2] – 18*U[1]^4*V[1]*V[2]^2 – 48*U[1]^3*U[2]^2*V[1]^2 – 56*U[1]^3*U[2]*V[1]^2*V[2] – 16*U[1]^3*U[3]*V[1]^3 – 48*U[1]^3*V[1]^2*V[2]^2 + 88*U[1]^2*U[2]^2*V[1]^3 + 40*U[1]^2*U[3]*V[1]^4 + 84*U[1]^2*V[1]^3*V[2]^2 – 40*U[1]*U[2]^2*V[1]^4 – 32*U[1]*U[3]*V[1]^5 – 30*U[1]*V[1]^4*V[2]^2 + 8*U[2]*V[1]^5*V[2] + 8*U[3]*V[1]^6 – 48*U[1]^4*U[3]*V[2] – 96*U[1]^4*U[4]*V[1] – 16*U[1]^3*U[2]^2*V[2] + 64*U[1]^3*U[2]*U[3]*V[1] – 56*U[1]^3*U[2]*V[2]^2 + 48*U[1]^3*U[3]*V[1]*V[2] + 288*U[1]^3*U[4]*V[1]^2 + 19*U[1]^3*V[2]^3 + 32*U[1]^2*U[2]^3*V[1] – 32*U[1]^2*U[2]^2*V[1]*V[2] – 64*U[1]^2*U[2]*U[3]*V[1]^2 + 186*U[1]^2*U[2]*V[1]*V[2]^2 – 36*U[1]^2*U[3]*V[1]^2*V[2] – 280*U[1]^2*U[4]*V[1]^3 – 27*U[1]^2*V[1]*V[2]^3 – 32*U[1]*U[2]^3*V[1]^2 – 32*U[1]*U[2]*U[3]*V[1]^3 – 90*U[1]*U[2]*V[1]^2*V[2]^2 + 60*U[1]*U[3]*V[1]^3*V[2] + 80*U[1]*U[4]*V[1]^4 – 20*U[1]*V[1]^2*V[2]^3 + 40*U[2]^2*V[1]^3*V[2] + 40*U[2]*U[3]*V[1]^4 – 24*U[2]*V[1]^3*V[2]^2 – 32*U[3]*V[1]^4*V[2] + 20*V[1]^3*V[2]^3 + 128*U[1]^3*U[2]*U[4] + 96*U[1]^3*U[4]*V[2] – 64*U[1]^2*U[2]^2*U[3] + 160*U[1]^2*U[2]*U[3]*V[2] – 320*U[1]^2*U[2]*U[4]*V[1] – 64*U[1]^2*U[3]^2*V[1] + 12*U[1]^2*U[3]*V[2]^2 – 384*U[1]^2*U[4]*V[1]*V[2] – 16*U[1]*U[2]^2*V[2]^2 + 16*U[1]*U[2]*U[3]*V[1]*V[2] + 224*U[1]*U[2]*U[4]*V[1]^2 – 40*U[1]*U[2]*V[2]^3 + 112*U[1]*U[3]^2*V[1]^2 – 120*U[1]*U[3]*V[1]*V[2]^2 + 480*U[1]*U[4]*V[1]^2*V[2] + 20*U[1]*V[2]^4 + 32*U[2]^3*V[1]*V[2] + 32*U[2]^2*U[3]*V[1]^2 – 48*U[2]^2*V[1]*V[2]^2 – 144*U[2]*U[3]*V[1]^2*V[2] + 72*U[2]*V[1]*V[2]^3 – 56*U[3]^2*V[1]^3 + 108*U[3]*V[1]^2*V[2]^2 – 160*U[4]*V[1]^3*V[2] – 20*V[1]*V[2]^4 – 128*U[1]^2*U[3]*U[4] + 128*U[1]*U[2]*U[3]^2 + 128*U[1]*U[2]*U[4]*V[2] – 272*U[1]*U[3]^2*V[2] + 256*U[1]*U[3]*U[4]*V[1] – 64*U[2]^2*U[3]*V[2] – 32*U[2]*U[3]^2*V[1] + 128*U[2]*U[3]*V[2]^2 – 256*U[2]*U[4]*V[1]*V[2] + 208*U[3]^2*V[1]*V[2] – 192*U[3]*U[4]*V[1]^2 – 64*U[3]*V[2]^3 – 64*U[3]^3 + 256*U[3]*U[4]*V[2]

[Po] [Th] [Be]

(x*y,x*z,x*u,y*z,y*u,z^2,z*u,u^2,x^3,y^3)

l0 = 6

(x^2, y^2)^c

Fundamental class = 12*U[1]^6*V[1]^2*V[2] – 8*U[1]^5*U[2]*V[1]^3 – 42*U[1]^5*V[1]^3*V[2] + 28*U[1]^4*U[2]*V[1]^4 + 54*U[1]^4*V[1]^4*V[2] – 36*U[1]^3*U[2]*V[1]^5 – 30*U[1]^3*V[1]^5*V[2] + 20*U[1]^2*U[2]*V[1]^6 + 6*U[1]^2*V[1]^6*V[2] – 4*U[1]*U[2]*V[1]^7 + 12*U[1]^6*V[2]^2 – 48*U[1]^5*U[2]*V[1]*V[2] – 30*U[1]^5*V[1]*V[2]^2 + 32*U[1]^4*U[2]^2*V[1]^2 + 112*U[1]^4*U[2]*V[1]^2*V[2] + 8*U[1]^4*U[3]*V[1]^3 + 51*U[1]^4*V[1]^2*V[2]^2 – 88*U[1]^3*U[2]^2*V[1]^3 – 76*U[1]^3*U[2]*V[1]^3*V[2] – 28*U[1]^3*U[3]*V[1]^4 – 75*U[1]^3*V[1]^3*V[2]^2 +
80*U[1]^2*U[2]^2*V[1]^4 + 12*U[1]^2*U[2]*V[1]^4*V[2] + 36*U[1]^2*U[3]*V[1]^5 + 57*U[1]^2*V[1]^4*V[2]^2 – 24*U[1]*U[2]^2*V[1]^5 – 4*U[1]*U[2]*V[1]^5*V[2] – 20*U[1]*U[3]*V[1]^6 – 15*U[1]*V[1]^5*V[2]^2 + 4*U[2]*V[1]^6*V[2] + 4*U[3]*V[1]^7 – 12*U[1]^5*U[3]*V[2] – 24*U[1]^5*U[4]*V[1] + 36*U[1]^4*U[2]^2*V[2] + 16*U[1]^4*U[2]*U[3]*V[1] – 56*U[1]^4*U[2]*V[2]^2 + 66*U[1]^4*U[3]*V[1]*V[2] + 60*U[1]^4*U[4]*V[1]^2 + 39*U[1]^4*V[2]^3 – 40*U[1]^3*U[2]^3*V[1] + 26*U[1]^3*U[2]^2*V[1]*V[2] – 96*U[1]^3*U[2]*U[3]*V[1]^2 – 14*U[1]^3*U[2]*V[1]*V[2]^2 – 70*U[1]^3*U[3]*V[1]^2*V[2] – 32*U[1]^3*U[4]*V[1]^3 – 60*U[1]^3*V[1]*V[2]^3 + 76*U[1]^2*U[2]^3*V[1]^2 – 114*U[1]^2*U[2]^2*V[1]^2*V[2] + 164*U[1]^2*U[2]*U[3]*V[1]^3 + 148*U[1]^2*U[2]*V[1]^2*V[2]^2 – 20*U[1]^2*U[3]*V[1]^3*V[2] – 20*U[1]^2*U[4]*V[1]^4 + 31*U[1]^2*V[1]^2*V[2]^3 – 36*U[1]*U[2]^3*V[1]^3 + 28*U[1]*U[2]^2*V[1]^3*V[2] – 108*U[1]*U[2]*U[3]*V[1]^4 – 66*U[1]*U[2]*V[1]^3*V[2]^2 + 52*U[1]*U[3]*V[1]^4*V[2] + 16*U[1]*U[4]*V[1]^5 – 20*U[1]*V[1]^3*V[2]^3 + 24*U[2]^2*V[1]^4*V[2] + 24*U[2]*U[3]*V[1]^5 – 12*U[2]*V[1]^4*V[2]^2 – 16*U[3]*V[1]^5*V[2] + 10*V[1]^4*V[2]^3 + 32*U[1]^4*U[2]*U[4] + 24*U[1]^4*U[4]*V[2] – 16*U[1]^3*U[2]^2*U[3] – 16*U[1]^3*U[2]*U[3]*V[2] – 16*U[1]^3*U[2]*U[4]*V[1] – 16*U[1]^3*U[3]^2*V[1] + 29*U[1]^3*U[3]*V[2]^2 – 186*U[1]^3*U[4]*V[1]*V[2] + 16*U[1]^2*U[2]^4 – 72*U[1]^2*U[2]^3*V[2] + 104*U[1]^2*U[2]^2*U[3]*V[1] + 149*U[1]^2*U[2]^2*V[2]^2 – 142*U[1]^2*U[2]*U[3]*V[1]*V[2] – 68*U[1]^2*U[2]*U[4]*V[1]^2 – 128*U[1]^2*U[2]*V[2]^3 + 64*U[1]^2*U[3]^2*V[1]^2 + 29*U[1]^2*U[3]*V[1]*V[2]^2 + 280*U[1]^2*U[4]*V[1]^2*V[2] + 41*U[1]^2*V[2]^4 – 16*U[1]*U[2]^4*V[1] + 32*U[1]*U[2]^3*V[1]*V[2] – 120*U[1]*U[2]^2*U[3]*V[1]^2 – 51*U[1]*U[2]^2*V[1]*V[2]^2 + 228*U[1]*U[2]*U[3]*V[1]^2*V[2] + 48*U[1]*U[2]*U[4]*V[1]^3 + 48*U[1]*U[2]*V[1]*V[2]^3 – 76*U[1]*U[3]^2*V[1]^3 – 88*U[1]*U[3]*V[1]^2*V[2]^2 – 80*U[1]*U[4]*V[1]^3*V[2] – 25*U[1]*V[1]*V[2]^4 + 36*U[2]^3*V[1]^2*V[2] + 36*U[2]^2*U[3]*V[1]^3 – 84*U[2]^2*V[1]^2*V[2]^2 – 80*U[2]*U[3]*V[1]^3*V[2] + 64*U[2]*V[1]^2*V[2]^3 + 28*U[3]^2*V[1]^4 + 36*U[3]*V[1]^3*V[2]^2 – 32*U[4]*V[1]^4*V[2] – 10*V[1]^2*V[2]^4 – 32*U[1]^3*U[3]*U[4] – 64*U[1]^2*U[2]^2*U[4] + 32*U[1]^2*U[2]*U[3]^2 + 120*U[1]^2*U[2]*U[4]*V[2] – 32*U[1]^2*U[3]^2*V[2] + 8*U[1]^2*U[3]*U[4]*V[1] + 46*U[1]^2*U[4]*V[2]^2 – 32*U[1]*U[2]^3*U[3] + 128*U[1]*U[2]^2*U[3]*V[2] + 80*U[1]*U[2]^2*U[4]*V[1] – 88*U[1]*U[2]*U[3]^2*V[1] – 170*U[1]*U[2]*U[3]*V[2]^2 – 64*U[1]*U[2]*U[4]*V[1]*V[2] + 110*U[1]*U[3]^2*V[1]*V[2] + 48*U[1]*U[3]*U[4]*V[1]^2 + 74*U[1]*U[3]*V[2]^3 – 260*U[1]*U[4]*V[1]*V[2]^2 + 16*U[2]^4*V[2] + 16*U[2]^3*U[3]*V[1] – 72*U[2]^3*V[2]^2 – 64*U[2]^2*U[3]*V[1]*V[2] + 114*U[2]^2*V[2]^3 + 44*U[2]*U[3]^2*V[1]^2 + 96*U[2]*U[3]*V[1]*V[2]^2 – 80*U[2]*U[4]*V[1]^2*V[2] – 76*U[2]*V[2]^4 – 64*U[3]^2*V[1]^2*V[2] – 16*U[3]*U[4]*V[1]^3 – 48*U[3]*V[1]*V[2]^3 + 208*U[4]*V[1]^2*V[2]^2 + 18*V[2]^5 + 32*U[1]^2*U[4]^2 + 96*U[1]*U[2]*U[3]*U[4] – 16*U[1]*U[3]^3 – 64*U[1]*U[3]*U[4]*V[2] – 64*U[1]*U[4]^2*V[1] + 16*U[2]^2*U[3]^2 – 96*U[2]^2*U[4]*V[2] – 40*U[2]*U[3]^2*V[2] – 64*U[2]*U[3]*U[4]*V[1] + 224*U[2]*U[4]*V[2]^2 + 24*U[3]^3*V[1] + 18*U[3]^2*V[2]^2 – 104*U[4]*V[2]^3 – 32*U[3]^2*U[4] + 128*U[4]^2*V[2]

[Po][Th][Be]

(x*z,x*u,y^2,y*z,y*u,z^2,z*u,u^2,y*x^2,x^3)

l0 = 6

(x^2, xy)^c

Fundamental class = 24*U[1]^6*V[1]^3*V[2] – 16*U[1]^5*U[2]*V[1]^4 – 84*U[1]^5*V[1]^4*V[2] + 56*U[1]^4*U[2]*V[1]^5 + 108*U[1]^4*V[1]^5*V[2] – 72*U[1]^3*U[2]*V[1]^6 – 60*U[1]^3*V[1]^6*V[2] + 40*U[1]^2*U[2]*V[1]^7 + 12*U[1]^2*V[1]^7*V[2] – 8*U[1]*U[2]*V[1]^8 – 24*U[1]^6*V[1]*V[2]^2 – 64*U[1]^5*U[2]*V[1]^2*V[2] + 108*U[1]^5*V[1]^2*V[2]^2 + 64*U[1]^4*U[2]^2*V[1]^3 + 112*U[1]^4*U[2]*V[1]^3*V[2] + 16*U[1]^4*U[3]*V[1]^4 – 114*U[1]^4*V[1]^3*V[2]^2 – 176*U[1]^3*U[2]^2*V[1]^4 – 8*U[1]^3*U[2]*V[1]^4*V[2] – 56*U[1]^3*U[3]*V[1]^5 – 30*U[1]^3*V[1]^4*V[2]^2 + 160*U[1]^2*U[2]^2*V[1]^5 – 56*U[1]^2*U[2]*V[1]^5*V[2] + 72*U[1]^2*U[3]*V[1]^6 + 90*U[1]^2*V[1]^5*V[2]^2 – 48*U[1]*U[2]^2*V[1]^6 + 8*U[1]*U[2]*V[1]^6*V[2] – 40*U[1]*U[3]*V[1]^7 – 30*U[1]*V[1]^6*V[2]^2 + 8*U[2]*V[1]^7*V[2] + 8*U[3]*V[1]^8 + 32*U[1]^5*U[2]*V[2]^2 + 72*U[1]^5*U[3]*V[1]*V[2] + 48*U[1]^5*U[4]*V[1]^2 – 16*U[1]^5*V[2]^3 + 40*U[1]^4*U[2]^2*V[1]*V[2] – 64*U[1]^4*U[2]*U[3]*V[1]^2 – 64*U[1]^4*U[2]*V[1]*V[2]^2 – 204*U[1]^4*U[3]*V[1]^2*V[2] – 184*U[1]^4*U[4]*V[1]^3 – 22*U[1]^4*V[1]*V[2]^3 – 80*U[1]^3*U[2]^3*V[1]^2 + 100*U[1]^3*U[2]^2*V[1]^2*V[2] + 112*U[1]^3*U[2]*U[3]*V[1]^3 – 220*U[1]^3*U[2]*V[1]^2*V[2]^2 + 292*U[1]^3*U[3]*V[1]^3*V[2] + 256*U[1]^3*U[4]*V[1]^4 + 188*U[1]^3*V[1]^2*V[2]^3 + 152*U[1]^2*U[2]^3*V[1]^3 – 228*U[1]^2*U[2]^2*V[1]^3*V[2] + 8*U[1]^2*U[2]*U[3]*V[1]^4 + 392*U[1]^2*U[2]*V[1]^3*V[2]^2 – 296*U[1]^2*U[3]*V[1]^4*V[2] – 152*U[1]^2*U[4]*V[1]^5 – 190*U[1]^2*V[1]^3*V[2]^3 – 72*U[1]*U[2]^3*V[1]^4 + 40*U[1]*U[2]^2*V[1]^4*V[2] – 104*U[1]*U[2]*U[3]*V[1]^5 – 100*U[1]*U[2]*V[1]^4*V[2]^2 + 184*U[1]*U[3]*V[1]^5*V[2] + 32*U[1]*U[4]*V[1]^6 + 20*U[1]*V[1]^4*V[2]^3 + 48*U[2]^2*V[1]^5*V[2] + 48*U[2]*U[3]*V[1]^6 – 40*U[2]*V[1]^5*V[2]^2 – 48*U[3]*V[1]^6*V[2] + 20*V[1]^5*V[2]^3 – 48*U[1]^5*U[4]*V[2] – 80*U[1]^4*U[2]*U[3]*V[2] – 96*U[1]^4*U[2]*U[4]*V[1] + 48*U[1]^4*U[3]*V[2]^2 + 168*U[1]^4*U[4]*V[1]*V[2] + 128*U[1]^3*U[2]^2*U[3]*V[1] – 80*U[1]^3*U[2]^2*V[2]^2 – 56*U[1]^3*U[2]*U[3]*V[1]*V[2] + 304*U[1]^3*U[2]*U[4]*V[1]^2 + 136*U[1]^3*U[2]*V[2]^3 + 64*U[1]^3*U[3]^2*V[1]^2 – 6*U[1]^3*U[3]*V[1]*V[2]^2 – 108*U[1]^3*U[4]*V[1]^2*V[2] – 36*U[1]^3*V[2]^4 + 32*U[1]^2*U[2]^4*V[1] – 112*U[1]^2*U[2]^3*V[1]*V[2] – 128*U[1]^2*U[2]^2*U[3]*V[1]^2 + 330*U[1]^2*U[2]^2*V[1]*V[2]^2 – 36*U[1]^2*U[2]*U[3]*V[1]^2*V[2] – 312*U[1]^2*U[2]*U[4]*V[1]^3 – 288*U[1]^2*U[2]*V[1]*V[2]^3 – 176*U[1]^2*U[3]^2*V[1]^3 + 42*U[1]^2*U[3]*V[1]^2*V[2]^2 – 160*U[1]^2*U[4]*V[1]^3*V[2] + 18*U[1]^2*V[1]*V[2]^4 – 32*U[1]*U[2]^4*V[1]^2 + 32*U[1]*U[2]^3*V[1]^2*V[2] – 64*U[1]*U[2]^2*U[3]*V[1]^3 – 70*U[1]*U[2]^2*V[1]^2*V[2]^2 + 408*U[1]*U[2]*U[3]*V[1]^3*V[2] + 96*U[1]*U[2]*U[4]*V[1]^4 – 40*U[1]*U[2]*V[1]^2*V[2]^3 + 168*U[1]*U[3]^2*V[1]^4 – 192*U[1]*U[3]*V[1]^3*V[2]^2 + 224*U[1]*U[4]*V[1]^4*V[2] + 90*U[1]*V[1]^2*V[2]^4 + 72*U[2]^3*V[1]^3*V[2] + 72*U[2]^2*U[3]*V[1]^4 – 152*U[2]^2*V[1]^3*V[2]^2 – 256*U[2]*U[3]*V[1]^4*V[2] + 160*U[2]*V[1]^3*V[2]^3 – 56*U[3]^2*V[1]^5 + 120*U[3]*V[1]^4*V[2]^2 – 64*U[4]*V[1]^5*V[2] – 60*V[1]^3*V[2]^4 + 64*U[1]^3*U[2]^2*U[4] + 96*U[1]^3*U[2]*U[4]*V[2] + 16*U[1]^3*U[3]^2*V[2] + 32*U[1]^3*U[3]*U[4]*V[1] – 204*U[1]^3*U[4]*V[2]^2 – 64*U[1]^2*U[2]^3*U[3] + 224*U[1]^2*U[2]^2*U[3]*V[2] – 192*U[1]^2*U[2]^2*U[4]*V[1] – 192*U[1]^2*U[2]*U[3]^2*V[1] – 180*U[1]^2*U[2]*U[3]*V[2]^2 – 336*U[1]^2*U[2]*U[4]*V[1]*V[2] + 312*U[1]^2*U[3]^2*V[1]*V[2] – 96*U[1]^2*U[3]*U[4]*V[1]^2 + 12*U[1]^2*U[3]*V[2]^3 + 516*U[1]^2*U[4]*V[1]*V[2]^2 + 128*U[1]*U[2]^2*U[3]*V[1]*V[2] + 160*U[1]*U[2]^2*U[4]*V[1]^2 – 84*U[1]*U[2]^2*V[2]^3 + 272*U[1]*U[2]*U[3]^2*V[1]^2 – 284*U[1]*U[2]*U[3]*V[1]*V[2]^2 + 384*U[1]*U[2]*U[4]*V[1]^2*V[2] + 120*U[1]*U[2]*V[2]^4 – 556*U[1]*U[3]^2*V[1]^2*V[2] + 96*U[1]*U[3]*U[4]*V[1]^3 + 196*U[1]*U[3]*V[1]*V[2]^3 – 456*U[1]*U[4]*V[1]^2*V[2]^2 – 36*U[1]*V[2]^5 + 32*U[2]^4*V[1]*V[2] + 32*U[2]^3*U[3]*V[1]^2 – 112*U[2]^3*V[1]*V[2]^2 – 272*U[2]^2*U[3]*V[1]^2*V[2] + 212*U[2]^2*V[1]*V[2]^3 – 88*U[2]*U[3]^2*V[1]^3 + 416*U[2]*U[3]*V[1]^2*V[2]^2 – 160*U[2]*U[4]*V[1]^3*V[2] – 168*U[2]*V[1]*V[2]^4 + 240*U[3]^2*V[1]^3*V[2] – 32*U[3]*U[4]*V[1]^4 – 208*U[3]*V[1]^2*V[2]^3 + 96*U[4]*V[1]^3*V[2]^2 + 36*V[1]*V[2]^5 – 64*U[1]^3*U[4]^2 – 96*U[1]^2*U[3]*U[4]*V[2] + 192*U[1]^2*U[4]^2*V[1] + 128*U[1]*U[2]^2*U[3]^2 + 64*U[1]*U[2]^2*U[4]*V[2] – 448*U[1]*U[2]*U[3]^2*V[2] + 64*U[1]*U[2]*U[3]*U[4]*V[1] + 128*U[1]*U[2]*U[4]*V[2]^2 + 64*U[1]*U[3]^3*V[1] + 276*U[1]*U[3]^2*V[2]^2 + 128*U[1]*U[3]*U[4]*V[1]*V[2] – 128*U[1]*U[4]^2*V[1]^2 – 112*U[1]*U[4]*V[2]^3 – 64*U[2]^3*U[3]*V[2] – 32*U[2]^2*U[3]^2*V[1] + 224*U[2]^2*U[3]*V[2]^2 – 192*U[2]^2*U[4]*V[1]*V[2] + 240*U[2]*U[3]^2*V[1]*V[2] – 128*U[2]*U[3]*U[4]*V[1]^2 – 256*U[2]*U[3]*V[2]^3 + 64*U[2]*U[4]*V[1]*V[2]^2 – 64*U[3]^3*V[1]^2 – 180*U[3]^2*V[1]*V[2]^2 + 64*U[3]*U[4]*V[1]^2*V[2] + 96*U[3]*V[2]^4 + 112*U[4]*V[1]*V[2]^3 – 64*U[1]*U[3]^2*U[4] – 256*U[1]*U[4]^2*V[2] – 64*U[2]*U[3]^3 + 256*U[2]*U[3]*U[4]*V[2] + 96*U[3]^3*V[2] + 64*U[3]^2*U[4]*V[1] – 384*U[3]*U[4]*V[2]^2 + 256*U[4]^2*V[1]*V[2]

$\Sigma^5$

$d=(5,1)$

(x^2-y^2,x^2-z^2,x^2-v^2,x^2-w^2,x*y,x*z,x*v,x*w,y*z,y*v,y*w,z*v,z*w,v*w)
l0=9

(x^2-y^2,x^2-z^2,x^2-v^2,w^2,x*y,x*z,x*v,x*w,y*z,y*v,y*w,z*v,z*w,v*w)
l0=9

(x^2-y^2,x^2-z^2,v^2,w^2,x*y,x*z,x*v,x*w,y*z,y*v,y*w,z*v,z*w,v*w)
l0=9

(x^2-y^2,z^2,v^2,w^2,x*y,x*z,x*v,x*w,y*z,y*v,y*w,z*v,z*w,v*w)
l0=9

(x^3,y^2,z^2,v^2,w^2,x*y,x*z,x*v,x*w,y*z,y*v,y*w,z*v,z*w,v*w)
l0=10

$\Sigma^6$

$d=(6)$

l0=(11 choose 5) – 6 = 456

List of 7 dimensional associative, commutative, local algebras over C

A6=(x^7) — codim=6l+6

I25=(xy,x^2+y^5) — codim=6l+7

III26=(x^2,xy,y^6) — codim=6l+8

I34=(xy,x^3+y^4) — codim=6l+7

III35=(xy,x^3,y^5) — codim=6l+8

(x^2-y^4,xy^2,y^5) — codim=6l+9

(x^2,xy^2,y^5) — codim=6l+10

III_{24}=(xy,x^4,y^4) — codim=6l+8

B6=(x^3+y^3,xy^2) — codim=6l+8

(x^2-y^3,xy^3,y^4) — codim=6l+9

(x^2,xy^3,y^4) — codim=6l+10

(x^3,x^2y-xy^2,y^3) — codim=6l+9

(x^3,x^2y,y^3) — codim=6l+10

(x^4,x^2y,xy^2,y^3) — codim=6l+11

(x^4-z^2,y^2-z^2,xy,xz,yz) — codim=6l+9

(x^4-z^2,y^2,xy,xz,yz) — codim=6l+10

(xy,xz,yz,y^2,z^2,x^5) — codim=6l+12

C(6,321,1)=(xy,xz,y^2-x^3,z^2-x^3) — codim=6l+8

C(6,321,2)=(xy,xz,y^2,z^2-x^3) — codim=6l+9

C(6,321,3)=(xy,y^2-xz,yz,z^2-x^3) — codim=6l+9

C(6,321,4)=(xy,xz,yz,y^3-z^3,x^2-z^3) — codim=6l+9

C(6,321,5)=(xy,xz,y^2,z^2,x^4) — codim=6l+10

C(6,321,6)=(x^2,xy,xz,yz,y^3-z^3) — codim=6l+10

C(6,321,7)=(xy,xz,yz,y^3,x^2-z^3) — codim=6l+10

C(6,321,8)=(xy,xz,y^2,z^3,x^2-yz^2) — codim=6l+10

C(6,321,9)=(xy,y^2-xz,z^2,yz-x^3,x^4) — codim=6l+10

C(6,321,10)=(x^2,xz,yz,y^3,xy-z^3) — codim=6l+11

C(6,321,11)=(x^2,xy,xz,y^2,z^3) — codim=6l+11

C(6,321,12)=(xy,y^2-xz,z^2,yz,x^4) — codim=6l+11

C(6,321,13)=(z^2,y^2,xy-xz,x^2z,x^3-yz) — codim=6l+11

C(6,321,14)=(xy,xz,yz,x^2,y^3,z^4) — codim=6l+12

C(6,321,15)=(y^2,xz,yz^2,xy,x^2-z^3,z^4) — codim=6l+12

C(6,321,16)=(x^2,y^2,xz,yz^2,xy-z^3,z^4) — codim=6l+12

C(6,321,17)=(x^2,xy,xz,y^2-z^3,yz^2,z^4) — codim=6l+13

C(6,321,18)=(x^2,y^2,xz,yz^2,xy,z^4) — codim=6l+14

NOC-A(c,g) , mu=(c:g^2) in P^1 — codim=6l+10 / 6l+9

NOC-C=(xy,z^2,y^2-2xz,x^3) — codim=6l+11

NOC-D=(xz,yz,z^2-2xy,x^3,y^3) — codim=6l+11

NOC-D*=(x^2,y^2,z^2-2xy,z^3) — codim=6l+11

NOC-E=(xy,xz,yz,x^3,y^3,z^3) — codim=6l+12

NOC-E*=(x^2,y^2,z^2,xyz) — codim=6l+12

NOC-F=(xz,z^2,x^2-y^2,x^3,x^2y) — codim=6l+12

NOC-F*=(x^2-y^2,xy,yz,z^3,xz^2) — codim=6l+12

NOC-G=(z^2,xy,xz,x^3,y^3,y^2z) — codim=6l+13

NOC-G*=(x^2,y^2,yz,z^3,xz^2) — codim=6l+13

NOC-H=(y^2-2xy,yz,z^2,x^3,x^2y,xy^2) — codim=6l+14

NOC-I=(xz,yz,z^2,x^3,x^2y,xy^2,y^3) — codim=6l+16

NOC-I*=(x^2,xy,y^2,xz^2,yz^2,z^3) — codim=6l+16

(x^3-y^2,x^3-z^2,x^3-u^2,xy,xz,xu,yz,yu,zu) — codim=6l+12

(x^3-y^2,x^3-z^2,u^2,xy,xz,xu,yz,yu,zu) — codim=6l+13

(x^3-y^2,z^2,u^2,xy,xz,xu,yz,yu,zu) — codim=6l+15

(xy,xz,xu,yz,yu,zu,y^2,z^2,x^4) — codim=6l+18

POQ-0*={1,1,1,1}=(xy,xz,xu,yz,yu,zu,z^2-x^2-y^2,u^2-x^2-cy^2), where c is a parameter — codim=6l+8 / 6l+9

POQ-1={2,1,1}=(x^2,xz,xu,yz,yu,zu,z^2-xy-y^2,u^2-xy+y^2) — codim=6l+9

POQ-2a={(1,1),1,1}=(xy,xz,xu,y^2,yz,yu,zu,z^2-u^2) — codim=6l+10

POQ-2b={3,1}=(x^2,xy,xu,yu,z^2,zu,y^2,yz-u^2) — codim=6l+10

POQ-2c={2,2}=(x^2,xz,xu,y^2,yu,z^2,u^2,yz-zu) — codim=6l+10

POQ-3a={(2,1),1}=(x^2,xz,xu,yz,yu,zu,z^2,y^2-u^2) — codim=6l+11

POQ-3b={4}=(x^2,xy,xz,y^2,zu,u^2,xu,z^2) — codim=6l+11

POQ-3c={2,(1,1)}=(x^2,xz,xu,yz,yu,z^2,u^2,y^2) — codim=6l+11

POQ-4a={(3,1)}=(x^2,xy,xu,yu,z^2,zu,y^2,u^2) — codim=6l+12

POQ-4b={(1,1),(1,1)}=(x^2,xz,xu,y^2,yz,yu,z^2,u^2) — codim=6l+12

POQ-5a={(1,1,1),1}=(x^2,xz,xu,y^2,yz,yu,zu,u^2-z^2) — codim=6l+13

POQ-5b={(2,2)}=(x^2,xz,xu,y^2,yz,yu,zu,u^2-z^2) — codim=6l+13

POQ-6a={(2,1,1)}=(x^2,xz,xu,yz,yu,z^2,u^2,y^2) — codim=6l+14

POQ-5c=(x^2,xy,xu,y^2,yu,z^2,zu,u^2) — codim=6l+13

POQ-5d=(xy,xz,xu,yz,yu,z^2,u^2,x^2+y^2-z^2) — codim=6l+13

POQ-6b=(x^2,xz,xu,yz,yu,zu,u^2,y^2-z^2) — codim=6l+14

POQ-7a=(x^2,xy,xu,yu,z^2,zu,u^2,y^2) — codim=6l+15

POQ-7b=(x^2,xz,xu,y^2,yz,yu,zu,u^2,z^3) — codim=6l+15

POQ-8=(x^2,xz,xu,yz,yu,zu,u^2,z^2-y^2) — codim=6l+16

POQ-9=(x^2,xu,y^2,yz,yu,z^2,zu,u^2) — codim=6l+17

POQ-10=(xy,xz,xu,yz,yu,z^2,zu,u^2,x^3,y^3) — codim=6l+18

POQ-11=(xz,xu,y^2,yz,yu,z^2,zu,u^2,yx^2,x^3) — codim=6l+19

(x^2-y^2,x^2-z^2,x^2-v^2,x^2-w^2,xy,xz,xv,xw,yz,yv,yw,zv,zw,vw) — codim=6l+16

(x^2-y^2,x^2-z^2,x^2-v^2,w^2,xy,xz,xv,xw,yz,yv,yw,zv,zw,vw) — codim=6l+17

(x^2-y^2,x^2-z^2,v^2,w^2,xy,xz,xv,xw,yz,yv,yw,zv,zw,vw) — codim=6l+19

(x^2-y^2,z^2,v^2,w^2,xy,xz,xv,xw,yz,yv,yw,zv,zw,vw) — codim=6l+22