Nodal surface

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 Varchenko (1983), which is better than the one by Miyaoka (1984).

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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			${\displaystyle {\tfrac {4}{9}}d(d-1)^{2}}$ (Miyaoka 1984)
d ≡ 0 (mod 3)	${\displaystyle {\tbinom {d}{2}}\lfloor {\tfrac {d}{2}}\rfloor +({\tfrac {d^{2}}{3}}-d+1)\lfloor {\tfrac {d-1}{2}}\rfloor }$	Escudero
d ≡ ±1 (mod 6)	${\displaystyle (5d^{3}-14d^{2}+13d-4)/12}$	Chmutov
d ≡ ±2 (mod 6)	${\displaystyle (5d^{3}-13d^{2}+16d-8)/12}$	Chmutov

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