Nodal surface

For nodal surfaces in physics and chemistry, see Node (physics).

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.

Degree Lower bound Surface achieving lower bound Upper bound
10Plane 0
21Conical surface1
34Cayley's nodal cubic surface4
416Kummer surface16
5 31 Togliatti surface 31 (Beauville)
665 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
11 425 480
12 600 Sarti surface 645
d (1/12)d(d  1)(5d  9) (Chmutov 1992)(4/9)d(d  1)2 (Miyaoka 1984)

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