Nodal surface

Template:For 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 Template:Harvtxt, which is better than the one by Template:Harvtxt.

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			$\frac{4}{9}d(d-1{)}^2$ Template:Harv
d ≡ 0 (mod 3)	$(\genfrac{}{}{0ex}{}{d}{2})\lfloor \frac{d}{2}\rfloor +(\frac{d^2}{3}-d+1)\lfloor \frac{d-1}{2}\rfloor$	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

• Algebraic surface

References

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