{{for|nodal surfaces in physics and chemistry|Node (physics)}} In algebraic geometry, a '''nodal surface''' is a surface in a (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" |- ! 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> \tfrac49 d (d-1)^2 </math> {{harv|Miyaoka|1984}} |- | ''d'' ≡ 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'' ≡ ±1 (mod 6) || <math> (5d^3 - 14d^2 + 13d - 4)/12 </math> || Chmutov || |- | ''d'' ≡ ±2 (mod 6) || <math> (5d^3 - 13d^2 + 16d - 8)/12 </math> || 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 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