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 , which is better than the one by . 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 d ≡ 0 (mod 3) Escudero d ≡ ±1 (mod 6) Chmutov d ≡ ±2 (mod 6) Chmutov
See also
Algebraic surface
References