An ordinary double point of a complex plane curve is a singular point with two smooth local branches
having distinct tangent lines. For a real plane
curve, an ordinary double point with two real branches is a crunode,
while one whose branches are nonreal complex conjugates
is an acnode. The necessary vanishing conditions for an
ordinary double point of a plane curve given by 
are
 |
(1)
|
where 
denotes a partial derivative. The Hessian
of 
must also be nonsingular at the point.
Let 
(or 
)
be a smooth immersion parametrizing a space
curve. A point 
,
where 
denotes the image of 
, is an ordinary double point of the space
curve if its preimage under 
consists of two values 
and 
, and the two tangent vectors 
and 
are noncollinear. The two local branches corresponding
to these preimages have distinct tangent
lines.
Ordinary double points of an algebraic surface given by 
satisfy the necessary conditions
 |
(2)
|
where 
denotes a partial derivative. The Hessian
of 
must also be nonsingular at the point. An algebraic
surface has at most finitely many ordinary double points. Let 
be the maximum possible number of ordinary double points
on an algebraic surface of degree 
in complex projective
space 
.
The values for 
,
2, ..., 6 and known bounds for larger 
are 0, 1, 4, 16, 31, 65, 
, 
, 
, 
, 
, 
, ... (OEIS A046001;
Chmutov 1992, Endraß 1995, Labs 2004, 2005).
The value 
was known to Kummer (1864, as cited by Chmutov 1992), while 
was proved by Beauville (1980) and 
by Jaffe and Ruberman (1997). For 
, the classical upper bound is
![mu(d)<=1/2[d(d-1)^2-3]](/images/equations/OrdinaryDoublePoint/NumberedEquation3.svg) |
(3)
|
(Labs 2005, eq. 1.3). Examples of algebraic surfaces with many ordinary double points are given in the following table.
 | nodes | surface |
| 3 | 4 | Cayley cubic |
| 4 | 16 | Kummer surface |
| 5 | 31 | dervish |
| 6 | 65 | Barth sextic |
| 7 | 99 | Labs septic |
| 8 | 168 | Endraß octic |
| 9 | 216 | Chmutov surface |
| 10 | 345 | Barth decic |
| 11 | 425 | Chmutov surface |
| 12 | 600 | Sarti dodecic |
In complex algebraic geometry, an ordinary double point is the analytic hypersurface germ
of a nondegenerate quadratic form. It is an isolated singularity of type 
, with Coxeter-Dynkin
diagram 
,
and is also called a "node" or "simple double point." In dimension 
it has local analytic equation
 |
(4)
|
Its normalized volume is 
for 
(Li 2018, Example 5.3).
(
)." Journées de géométrie
algébrique d'Angers (1979). Sijthoff & Noordhoff, pp. 207-215,
1980.Chmutov, S. V. "Examples of Projective Surfaces with
Many Singularities." J. Algebraic Geom. 1, 191-196, 1992.Endraß,
S. "Flächen mit vielen Doppelpunkten." DMV-Mitteilungen 4,
17-20, Apr. 1995.Endraß, S. "Surfaces with Many Ordinary
Nodes."