Dear all,
in some of my work I am considering the algebraic surface given implicitly by
$$ z^2\left(1x+\frac{z^2}{4}\right) = y^2. $$
Maybe this surface is known? Does anyone recognize it and/or point me to some literature?
Thanks a lot!
Dear all, in some of my work I am considering the algebraic surface given implicitly by $$ z^2\left(1x+\frac{z^2}{4}\right) = y^2. $$ Maybe this surface is known? Does anyone recognize it and/or point me to some literature? Thanks a lot! 


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 selfintersecting 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 blowingup 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 blowup: it shows that blowingup the worse singularity is not the best way to smooth a space (if you blowup 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 Ftheory) when one consider "Sen's weak coupling limit" which gives the orientifold limit of Ftheory at weak coupling. 

