%I #27 Sep 16 2024 11:18:52
%S 0,1,4,16,31,65
%N Maximal number of ordinary double points on an n-th degree algebraic surface in complex projective 3-space.
%C For n >= 7 lower bounds are 99, 168, 226, 345, 425, 600, 732, 949; upper bounds are 104, 174, 246, 360, 480, 645, 829, 1051. The bounds on mu(d), the maximum number of singularities on a surface of degree d in three-space, are 5/12*d^3 <= mu(d) <= 4/9*d^3. - Corrected and extended by _Martin Renner_, Nov 10 2014
%H S. Endrass, <a href="https://web.archive.org/web/20031222231427/http://enriques.mathematik.uni-mainz.de/kon/docs/Eflaechen.shtml">Surfaces with many ordinary nodes</a>
%H S. Endrass, <a href="https://doi.org/10.1515/dmvm-2008-0090">Flaechen mit vielen Doppelpunkten</a> DMV-Mitteilungen 4 (April 1995), 17-20.
%H Soliver Labs, <a href="https://arxiv.org/abs/math/0409348">A septic with 99 real nodes</a>, arXiv:math/0409348 [math.AG] (2004).
%H Oliver Labs, <a href="https://web.archive.org/web/20170414105652/http://www.oliverlabs.net/view.php?menuitem=160">Nodal surfaces</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OrdinaryDoublePoint.html">Ordinary Double Point</a>
%e For n = 3 there is a unique surface of degree 3 with 4 double points, Cayley's cubic: 4(w^3+x^3+y^3+z^3) = (w+x+y+z)^3.
%K nonn,nice,hard,more
%O 1,3
%A _Eric W. Weisstein_