Singularities on Cubic Surfaces
The normal forms of the singularities that occur on cubic surfaces:
The 21 types of cubic surfaces in reference to isolated singularities
are listed below; click on the links to get different affine and real
views of the surfaces.
On the page on A, D, E - Singularities on cubic
surfaces, you can modify the Coxeter-Diagram to get the
corresponding cubic surface.
On the page on lines on cubic surfaces we
present one real view of each surface and the lines on them.
- I=12 (no singularity)
- II=12-C2 (one ordinary/conic double point)
- III=12-B3
(one biplanar double point B3)
- IV=12 - 2C2 (two ordinary/conic double points)
- V=12
- B4 (one biplanar double point B4)
- VI=12-
B3- C2 (one biplanar double point
B3 and an ordinary double point)
- VII=12-B5
(one biplanar double point B5)
- VIII=12-3C2 (three ordinary double points)
- IX=12-2 B3 (two biplanar double points B3)
- X=12-B4-C2
(one biplanar double point
B4 and an ordinary double point)
- XI=12-B6 (one biplanar double point B6)
- XII=12-U6 (one uniplanar double point U6)
- XIII=12-B3-2 C2 (one biplanar double point
B3 and two ordinary double points)
- XIV=12-B5-C2 (one biplanar double point
B5 and an ordinary double point)
- XV=12-U7
(one uniplanar double point U7)
- XVI=12-4
C2 (four ordinary double points)
- XVII=12-2B3-C2 (two biplanar double points
B3 and an ordinary double point)
- XVIII=12-B4-2 C2 (one biplanar double point
B4 and two ordinary double points)
- XIX=12-B6-C2 (one biplanar double point
B6 and an ordinary double point)
- XX=12-U8
(one uniplanar double point U8)
- XXI=12-3
B3 (three biplanar double points B3)
Author: Oliver Labs
at the department of mathematics
of the Gutenberg University of Mainz, Germany
Home: The Cubic Surface Homepage
of the algebraic geometry group
