Ordinary Double Point

An ordinary double point of a plane curve is point where a curve intersects itself such that two branches of the curve have distinct tangent lines. Ordinary double points of plane curves are commonly known as crunodes. Ordinary double points of a plane curves given by satisfy

(1)

where denotes a partial derivative.

Let (or ) be a space curve. Then a point (where denotes the immersion of ) is an ordinary double point of the space curve if its preimage under consists of two values and , and the two tangent vectors and are noncollinear. Geometrically, this means that, in a neighborhood of , the curve consists of two transverse branches. Ordinary double points are isolated singularities having Coxeter-Dynkin diagram of type , and also called "nodes" or "simple double points."

Ordinary double points of a surface given by satisfy

(2)

where denotes a partial derivative. A surface in complex three-space admits at most finitely many ordinary double points. The maximum possible number of ordinary double points for a surface of degree , 2, ..., are 0, 1, 4, 16, 31, 65, , , , , , ... (OEIS A046001; Chmutov 1992, Endraß 1995, Labs 2004).

was known to Kummer in 1864 (Chmutov 1992), the fact that was proved by Beauville (1980), and was proved by Jaffe and Ruberman (1997). For , the following inequality holds:

(3)

(Endraß 1995). Examples of algebraic surfaces having the maximum (known) number of ordinary double points are given in the following table.

		surface
3	4	Cayley cubic
4	16	Kummer surface
5	31	dervish
6	65	Barth sextic
7	99	Labs septic
8	168	Endraß octic
9	216	Chmutov surface
10	345	Barth decic
11	425	Chmutov surface
12	600	Sarti dodecic