ART

In algebraic geometry, a nodal surface is a surface in (usually complex) projective space whose only singularities are nodes. A major problem about them is to find the maximum number of nodes of a nodal surface of given degree.

The following table gives some known upper and lower bounds for the maximal number of nodes on a complex surface of given degree.

Degree Lower bound Surface achieving lower bound Upper bound
1 0 Plane 0
2 1 Conical surface 1
3 4 Cayley's nodal cubic surface 4
4 16 Kummer surface 16
5 31 Togliatti surface 31 (Beauville)
6 65 Barth sextic 65 (Jaffe and Ruberman)
7 99 Labs septic 104
8 168 Endraß surface 174
9 226 Labs 246
10 345 Barth decic 360
11 425 480
12 600 Sarti surface 645
d (1/12)d(d − 1)(5d − 9) (Chmutov 1992) (4/9)d(d − 1)2 (Miyaoka 1984)

See also

Algebraic surface

References
Chmutov, S. V. (1992), "Examples of projective surfaces with many singularities.", J. Algebraic Geom., 1 (2): 191–196, MR 1144435
Miyaoka, Yoichi (1984), "The maximal Number of Quotient Singularities on Surfaces with Given Numerical Invariants", Mathematische Annalen, 268 (2): 159–171, doi:10.1007/bf01456083

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License