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"
DegreeLower boundSurface achieving lower boundUpper bound
10Plane0
21Conical surface1
34Cayley's nodal cubic surface4
416Kummer surface16
531Togliatti surface31 (Beauville)
665Barth sextic65 (Jaffe and Ruberman)
799Labs septic104
8168Endraß surface174
9226Labs246
10345Barth decic360
11425Chmutov480
12600Sarti surface645
13732Chmutov829
d \tfrac49 d (d-1)^2 {{harv|Miyaoka|1984}}
d ≡ 0 (mod 3) \tbinom d2 \lfloor \tfrac d2 \rfloor + (\tfrac{d^2}3 - d + 1)\lfloor\tfrac{d-1}2\rfloor Escudero
d ≡ ±1 (mod 6) (5d^3 - 14d^2 + 13d - 4)/12 Chmutov
d ≡ ±2 (mod 6) (5d^3 - 13d^2 + 16d - 8)/12 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 }}

Category:Singularity theory

Category:Algebraic surfaces