## Ordinary Double Point

A Rational Double Point of Conic Double Point type, known as .'' An ordinary Double Point is called a Node. The above plot shows the curve , 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 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 inequality holds:

(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 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

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Beauville, A. Sur le nombre maximum de points doubles d'une surface dans ().'' 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.

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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.