Ordinary Double Point
A Rational Double Point of Conic Double Point type, known as ``
.'' An ordinary Double Pointis called a Node. The above plot shows the curve 
, which has anordinary double point at the Origin.
A surface in complex 3-space admits at most finitely many ordinary double points. The maximum possible number ofordinary double points 
for a surface of degree 
, 2, ..., are 0, 1, 4, 16, 31, 65, 
, 
, 
, 
, 
,
... (Sloane's A046001; Chmutov 1992, Endraß 1995). The fact that 
was proved by Beauville (1980), and 
was proved by Jaffe and Ruberman (1994). For 
, the following inequalityholds:
(Endraß 1995). Examples of Algebraic Surfaces having the maximum (known) number ofordinary double points are given in the following table.
 | | Surface |
| 3 | 4 | Cayley Cubic |
| 4 | 16 | Kummer Surface |
| 5 | 31 | Dervish |
| 6 | 65 | Barth Sextic |
| 8 | 168 | Endraß Octic |
| 10 | 345 | Barth Decic |
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 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 A046001in ``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.
().'' Journées de géométrie algébrique d'Angers (1979). Sijthoff & Noordhoff, pp. 207-215, 1980.
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