Nodal surface

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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 [math]\displaystyle{ \tfrac49 d (d-1)^2 }[/math] (Miyaoka 1984)
d ≡ 0 (mod 3) [math]\displaystyle{ \tbinom d2 \lfloor \tfrac d2 \rfloor + (\tfrac{d^2}3 - d + 1)\lfloor\tfrac{d-1}2\rfloor }[/math] Escudero
d ≡ ±1 (mod 6) [math]\displaystyle{ (5d^3 - 14d^2 + 13d - 4)/12 }[/math] Chmutov
d ≡ ±2 (mod 6) [math]\displaystyle{ (5d^3 - 13d^2 + 16d - 8)/12 }[/math] Chmutov

See also

References

  • 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 
  • 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 
  • Chmutov, S. V. (1992), "Examples of projective surfaces with many singularities.", J. Algebraic Geom. 1 (2): 191–196 
  • Escudero, Juan García (2013), "On a family of complex algebraic surfaces of degree 3n", C. R. Math. Acad. Sci. Paris 351 (17–18): 699–702, doi:10.1016/j.crma.2013.09.009