2004/12/02 | 常二重点[Ordinary Double Point]
类别(∑〖数学〗) | 评论(0) | 阅读(52) | 发表于 13:13

Let (or ) be a space curve. Then a point (where denotes the immersion of f) is an ordinary double point if its preimage under f consists of two values and , and the two tangent vectors and are noncollinear. Geometrically, this means that, in a neighborhood of p, 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."

The above plot shows the curve , which has an ordinary double point at the origin.
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 d = 1,2, ..., are 0, 1, 4, 16, 31, 65,,,,,,... (Sloane's A046001;Chmutov 1992, Endraß 1995).
was known to Kummer in 1864 (Chmutov 1992), the fact that was proved by Beauville (1980), and was proved by Jaffe and Ruberman (1994). For , the following inequality holds:

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

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