Nodal surface explained

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, which is better than the one by .

Degree	Lower bound	Surface achieving lower bound	Upper bound
1	0	Plane	0
2	1		1
3	4		4
4	16		16
5	31		31 (Beauville)
6	65		65 (Jaffe and Ruberman)
7	99		104
8	168		174
9	226	Labs	246
10	345		360
11	425	Chmutov	480
12	600		645
13	732	Chmutov	829
d			\tfrac 49 d (d -1 ) ^{2}
d ≡ 0 (mod 3)	\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)	( 5 d ^{3} - 14 d ^{2} + 13 d - 4 )/ 12	Chmutov
d ≡ ±2 (mod 6)	( 5 d ^{3} - 13 d ^{2} + 16 d - 8 )/ 12	Chmutov

See also

• Algebraic surface