Ordinary Double Point

A Rational Double Point of Conic Double Point type, known as ``$A_1$.'' An ordinary Double Point is called a Node. The above plot shows the curve $x^3-x^2+y^2=0$, which has an ordinary double point at the Origin.

A surface in complex 3-space admits at most finitely many ordinary double points. The maximum possible number of ordinary double points $\mu(d)$ for a surface of degree $d=1$, 2, ..., are 0, 1, 4, 16, 31, 65, $93\leq\mu(7)\leq 104$, $168\leq\mu(8)\leq 174$, $216\leq\mu(8)\leq 246$, $345\leq\mu(10)\leq 360$, $425\leq\mu(11)\leq 480$, $576\leq\mu(12)\leq 645$ ... (Sloane's A046001; Chmutov 1992, Endraß 1995). The fact that $\mu(5)=31$ was proved by Beauville (1980), and $\mu(6)=65$ was proved by Jaffe and Ruberman (1994). For $d\geq 3$, the following inequality holds:

$$\mu(d)\leq {\textstyle{1\over 2}}[d(d-1)-3]$$

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

$d$	$\mu(d)$	Surface
3	4	Cayley Cubic
4	16	Kummer Surface
5	31	Dervish
6	65	Barth Sextic
8	168	Endraß Octic
10	345	Barth Decic

See also Algebraic Surface, Barth Decic, Barth Sextic, Cayley Cubic, Cusp, Dervish, Endraß Octic, Kummer Surface, Rational Double Point

References

Basset, A. B. ``The Maximum Number of Double Points on a Surface.'' Nature 73, 246, 1906.

Beauville, A. ``Sur le nombre maximum de points doubles d'une surface dans $\Bbb{P}^3$ ($\mu(5)=31$).'' Journées de géométrie algébrique d'Angers (1979). Sijthoff & Noordhoff, pp. 207-215, 1980.

Chmutov, S. V. ``Examples of Projective Surfaces with Many Singularities.'' J. Algebraic Geom. 1, 191-196, 1992.

Endraß, S. ``Surfaces with Many Ordinary Nodes.'' http://www.mathematik.uni-mainz.de/AlgebraischeGeometrie/docs/Eflaechen.shtml.

Endraß, S. ``Flächen mit vielen Doppelpunkten.'' DMV-Mitteilungen 4, 17-20, Apr. 1995.

Endraß, S. Symmetrische Fläche mit vielen gewöhnlichen Doppelpunkten. Ph.D. thesis. Erlangen, Germany, 1996.

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

Jaffe, D. B. and Ruberman, D. ``A Sextic Surface Cannot have 66 Nodes.'' J. Algebraic Geom. 6, 151-168, 1997.

Miyaoka, Y. ``The Maximal Number of Quotient Singularities on Surfaces with Given Numerical Invariants.'' Math. Ann. 268, 159-171, 1984.

Sloane, N. J. A. Sequence A046001 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Togliatti, E. G. ``Sulle superficie algebriche col massimo numero di punti doppi.'' Rend. Sem. Mat. Torino 9, 47-59, 1950.

Varchenko, A. N. ``On the Semicontinuity of Spectrum and an Upper Bound for the Number of Singular Points on a Projective Hypersurface.'' Dokl. Acad. Nauk SSSR 270, 1309-1312, 1983.

Walker, R. J. Algebraic Curves. New York: Springer-Verlag, pp. 56-57, 1978.