Basic Alg. Geometry: Nodal Singularities. Attn Wonk.

[WWGD]

Hi, everyone :

I am reading about "Nodal singularities" ,and I have not been able to find
a clear def. of what they are. Neither of the standard sources: Wiki, Google,
or some of the books I have checked, has a clear explanation. Here is the
context:

We are trying to show that we can represent elements of H^2(M;Z), in a 4-D,
orientable manifold M, by embedded submanifolds.

An example mentioned is that of the coordinate planes Z1=0 and Z2=0 , meeing
at the origin. I guess these are the coordinate axes(planes) in C^2 ~ R^4 (C=complexes)
Now, mention is made that " a double-point is isomorphic to the simple nodal
singularity :

Z1*Z2=0

meeting at the origin. Now, I have an idea of what is going on, but I would like
to have actual definitions of "double-points" (equiv: n-point ) singularity, and
of nodal singularity. I believe that the singularity has to see with having a corner,
a sharp corner that is not smooth, tho I am not clear on what the issue is with
nodal singularity.

Also, it is mentioned that the singularities can be perturbed away slightly, at the
price of increasing the genus. Is this the standard def. of genus, i.e., the genus
of a surface is twice the number of non-separating loops on the surface ( or, equiv.
, I guess, non-separating manifolds?)

Thanks For Any Help.

 

[pat_connell]

A nodal singularity is a type of singularity in which the surface has a single point with two or more distinct tangent directions. This can occur when two curves intersect at a sharp corner, for example. A double-point is an example of a nodal singularity, which is when two curves intersect at a single point. The genus of a surface is the number of holes or handles that it has; it is related to the number of non-separating loops on the surface.