Question

Dear all,

in some of my work I am considering the algebraic surface given implicitly by

$$ z^2\left(1-x+\frac{z^2}{4}\right) = y^2. $$

Maybe this surface is known? Does anyone recognize it and/or point me to some literature?

Thanks a lot!

Answer 1

Your surface is actually a Whitney umbrella.

To see that, just perform the following substitution (which is just a translation):

$$x =1+\frac{z^2}{4} -t.$$

After this, your surface is defined by the simpler equation:

$$z^2 t =y^2.$$

This is exactly the canonical form of the Whitney umbrella. The Whitney umbrella is a singular surface in $\mathbb{C}^3$ which looks like a self-intersecting plane. It has a line of double points at $z=y=0$ and the singularity worsen to a pinch point at the origin $z=y=t=0$. You can resolve it by blowing-up the double line.

The Whitney umbrella is studied in many books of algebraic geometry as an example of a surface with a pinch point and a first tricky example of blow-up: it shows that blowing-up the worse singularity is not the best way to smooth a space (if you blow-up the pinch points you will again get the same equation). In classical singular theory, it is an example of a singular surface which does not have a regular Whitney stratification. It also appears very naturally in the study of elliptic fibration in the context of string theory (more precisely F-theory) when one consider "Sen's weak coupling limit" which gives the orientifold limit of F-theory at weak coupling.