du Val singularity

In

canonical singularities (or, equivalently, rational Gorenstein singularities) in dimension 2. They were studied by Patrick du Val^[1]^[2]^[3] and Felix Klein

.

The Du Val singularities also appear as quotients of $\mathbb {C} ^{2}$ by a finite subgroup of SL₂ $(\mathbb {C} )$ ; equivalently, a finite subgroup of

binary polyhedral groups.^[4] The rings of invariant polynomials of these finite group actions were computed by Klein, and are essentially the coordinate rings of the singularities; this is a classic result in invariant theory.^[5]^[6]

Classification

simply laced Dynkin diagrams, a form of ADE classification

.

The possible Du Val singularities are (up to analytical isomorphism):

$A_{n}:\quad w^{2}+x^{2}+y^{n+1}=0$
$D_{n}:\quad w^{2}+y(x^{2}+y^{n-2})=0\qquad (n\geq 4)$
$E_{6}:\quad w^{2}+x^{3}+y^{4}=0$
$E_{7}:\quad w^{2}+x(x^{2}+y^{3})=0$
$E_{8}:\quad w^{2}+x^{3}+y^{5}=0.$

References

S2CID 251095858. Archived from the original
on 9 May 2022.

S2CID 197459819. Archived from the original
on 13 May 2022.

S2CID 251095521. Archived from the original
on 9 May 2022.

OCLC 642357691. Archived
from the original on 2022-05-09. Retrieved 2022-05-09.

MR 0199191
.

MR 0543555. Archived
from the original on 2022-05-09. Retrieved 2022-05-09.

External links

Reid, Miles, The Du Val singularities A_n, D_n, E₆, E₇, E₈ (PDF)

Burban, Igor, Du Val Singularities (PDF)

Retrieved from "https://en.wikipedia.org/w/index.php?title=Du_Val_singularity&oldid=1145774025"

[1] S2CID 251095858. Archived from the original
on 9 May 2022.

[2] S2CID 197459819. Archived from the original
on 13 May 2022.

[3] S2CID 251095521. Archived from the original
on 9 May 2022.

[4] OCLC 642357691. Archived
from the original on 2022-05-09. Retrieved 2022-05-09.

[5] MR 0199191
.

[6] MR 0543555. Archived
from the original on 2022-05-09. Retrieved 2022-05-09.

[1]

[2]

[3]

[4]

[5]

[6]

Classification

See also

References

External links