Nodal surface
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. In degree 7, 9, 11, and 13, the upper bound is given by Varchenko (1983), which is better than the one by Miyaoka (1984).
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 | Chmutov | 480 |
12 | 600 | Sarti surface | 645 |
13 | 732 | Chmutov | 829 |
d | (1/12)d(d − 1)(5d − 9) | (Chmutov 1992) | (4/9)d(d − 1)2 (Miyaoka 1984) |
See also[]
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
- Varchenko, A. N. (1983), "Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface", Doklady Akademii Nauk SSSR, 270 (6): 1294–1297, MR 0712934
External links[]
- Labs, O., Nodal surfaces
Categories:
- Singularity theory
- Algebraic surfaces