- References
- External links
{{for|nodal surfaces in physics and chemistry|Node (physics)}}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}} | d(d − 1)2 {{harv>Miyaoka|1984}} |
References
|last=Chmutov|first= S. V.
|title=Examples of projective surfaces with many singularities.
|journal=J. Algebraic Geom. |volume=1 |year=1992|issue= 2|pages= 191–196}}- {{Citation | last1=Miyaoka | first1=Yoichi | title=The maximal Number of Quotient Singularities on Surfaces with Given Numerical Invariants | year=1984 | journal=Mathematische Annalen | volume=268 | issue=2 | pages=159–171 | doi=10.1007/bf01456083}}
External links- {{citation|first=O. |last=Labs|title=Nodal surfaces|url=http://www.oliverlabs.net/view.php?menuitem=160}}
2 : Singularity theory|Algebraic surfaces |