Nodal surface
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For nodal surfaces in physics and chemistry, see 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 Varchenko (1983), which is better than the one by Miyaoka (1984).
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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 | (Miyaoka 1984) | ||
d ≡ 0 (mod 3) | Escudero | ||
d ≡ ±1 (mod 6) | Chmutov | ||
d ≡ ±2 (mod 6) | Chmutov |
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