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).
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, MR 0712934
- 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, MR 0744605
- Chmutov, S. V. (1992), "Examples of projective surfaces with many singularities.", J. Algebraic Geom., 1 (2): 191–196, MR 1144435
- 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, arXiv:1302.6747, doi:10.1016/j.crma.2013.09.009, MR 3124329