Isolated Singularity

An isolated singularity is a Singularity for which there exists a (small) Real Number $\epsilon$ such that there are no other Singularities within a Neighborhood of radius $\epsilon$ centered about the Singularity.

The types of isolated singularities possible for Cubic Surfaces have been classified (Schläfli 1864, Cayley 1869, Bruce and Wall 1979) and are summarized in the following table from Fischer (1986).

Name	Symbol	Normal Form	Coxeter Diagram
conic double point	$C_2$	$x^2+y^2+z^2$	$A_1$
biplanar double point	$B_3$	$x^2+y^2+z^3$	$A_2$
biplanar double point	$B_4$	$x^2+y^2+z^4$	$A_3$
biplanar double point	$B_5$	$x^2+y^2+z^5$	$A_4$
biplanar double point	$B_6$	$x^2+y^2+z^6$	$A_5$
uniplanar double point	$U_6$	$x^2+z(y^2+z^2)$	$D_4$
uniplanar double point	$U_7$	$x^2+z(y^2+z^3)$	$D_5$
uniplanar double point	$U_8$	$x^2+y^3+z^4$	$E_6$
elliptic cone point	--	$xy^2-4z^3-g_2x^2y+g_3x^3$	$\tilde E_6$

See also Cubic Surface, Rational Double Point, Singularity

References

Bruce, J. and Wall, C. T. C. ``On the Classification of Cubic Surfaces.'' J. London Math. Soc. 19, 245-256, 1979.

Cayley, A. ``A Memoir on Cubic Surfaces.'' Phil. Trans. Roy. Soc. 159, 231-326, 1869.

Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 12-13, 1986.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 380-381, 1953.

Schläfli, L. ``On the Distribution of Surfaces of Third Order into Species.'' Phil. Trans. Roy. Soc. 153, 193-247, 1864.