# Nodal surface

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{{for|nodal surfaces in physics and chemistry|Node (physics)}}
In [algebraic geometry](/source/algebraic_geometry), a '''nodal surface''' is a surface in a (usually [complex](/source/Complex_projective_space)) [projective space](/source/projective_space) whose only singularities are [nodes](/source/Singular_point_of_an_algebraic_variety). 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](/source/Conical_surface) || 1
|-
| 3 || 4 || [Cayley's nodal cubic surface](/source/Cayley's_nodal_cubic_surface) || 4
|-
| 4 || 16 || [Kummer surface](/source/Kummer_surface) || 16
|-
| 5 || 31 || [Togliatti surface](/source/Togliatti_surface) || 31 (Beauville)
|-
| 6 || 65  || [Barth sextic](/source/Barth_sextic) || 65 (Jaffe and Ruberman)
|-
| 7 || 99 || [Labs septic](/source/Labs_septic) || 104
|-
| 8 || 168 || [Endraß surface](/source/Endra%C3%9F_surface) || 174
|-
| 9 || 226 || Labs || 246
|-
| 10 || 345  || [Barth decic](/source/Barth_decic) || 360
|-
| 11 || 425  || Chmutov  || 480
|-
| 12 || 600 || [Sarti surface](/source/Sarti_surface) || 645
|-
| 13 || 732 || Chmutov || 829
|-
| ''d'' || || || <math> \tfrac49 d (d-1)^2 </math> {{harv|Miyaoka|1984}}
|-
| ''d'' &equiv; 0 (mod 3) || <math> \tbinom d2 \lfloor \tfrac d2 \rfloor + (\tfrac{d^2}3 - d + 1)\lfloor\tfrac{d-1}2\rfloor </math> || Escudero ||
|-
| ''d'' &equiv; &plusmn;1 (mod 6) || <math> (5d^3 - 14d^2 + 13d - 4)/12 </math> || Chmutov ||
|-
| ''d'' &equiv; &plusmn;2 (mod 6) || <math> (5d^3 - 13d^2 + 16d - 8)/12 </math> || Chmutov ||
|}

==See also==

*[Algebraic surface](/source/Algebraic_surface)

==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 3''n'' | 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

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