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 Studies in Mathematics
Hellenica World - Scientific Library
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