nodal surface
{{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. In degree 7, 9, 11, and 13, the upper bound is given by {{harvtxt|Varchenko|1983}}, which is better than the one by {{harvtxt|Miyaoka|1984}}.
| class="wikitable" | |||
| 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 | {{harv|Miyaoka|1984}} | ||
| d ≡ 0 (mod 3) | Escudero | ||
| d ≡ ±1 (mod 6) | Chmutov | ||
| d ≡ ±2 (mod 6) | Chmutov |
See also
References
- {{citation | last = Varchenko | first = A. N. | authorlink = Alexander Varchenko | issue = 6 | journal = Doklady Akademii Nauk SSSR | mr = 712934 | pages = 1294–1297 | title = Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface | volume = 270 | year = 1983}}
- {{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 | mr=0744605 }}
- {{citation|mr=1144435
|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 | mr=3124329 | doi=10.1016/j.crma.2013.09.009 | last=Escudero | first=Juan García | title=On a family of complex algebraic surfaces of degree 3n | journal=C. R. Math. Acad. Sci. Paris | volume=351 | year=2013 | issue=17–18 | pages=699–702| arxiv=1302.6747 }}