# Basic Alg. Geometry: Nodal Singularities. Attn Wonk.

• WWGD
In summary: Singularities can be perturbed away, but this may increase the genus of the surface. In summary, nodal singularities are a type of singularity where two or more curves intersect at a single point, and can be perturbed away at the cost of increasing the genus of the surface.
WWGD
Gold Member
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.

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.

## 1. What is basic algebraic geometry?

Basic algebraic geometry is the study of geometric shapes and their relations using algebraic equations. It combines concepts from algebra and geometry to study the structure and properties of these shapes.

## 2. What are nodal singularities in algebraic geometry?

Nodal singularities are points on a curve or surface where the curve or surface is not smooth. This means that at these points, the tangent line or plane does not exist, and the curve or surface appears to have a sharp point or cusp.

## 3. How do nodal singularities differ from other types of singularities?

Nodal singularities are classified as isolated singularities, meaning they occur at isolated points on a curve or surface. This is different from non-isolated singularities, which occur over a larger region of the curve or surface.

## 4. What is the importance of studying nodal singularities in algebraic geometry?

Studying nodal singularities is important because they can provide important insights into the global structure of a curve or surface. Additionally, they have applications in areas such as computer graphics, where nodal singularities can be used to create realistic and detailed images.

## 5. Are there any real-world examples of nodal singularities?

Yes, there are many real-world examples of nodal singularities. For instance, a common example is the intersection of two cones, where the point of intersection is a nodal singularity. Other examples include the cusps on a heart-shaped curve and the singularities on a cactus plant.

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