Euler characteristic of the zero set of $z_1^d+z_2^d+z_3^d=0$ in $CP^2$

Question

I'm doing this qualifying exam problem from my university and got stuck with this one:

Let $Z$ be the zero set of the polynomial $z_1^d+z_2^d+z_3^d=0$ in $CP^2$, here $z_1,z_2,z_3$ are complex numbers; what is the Euler characteristic of $Z$?

I really have no clue what to do with this. I know that the Euler characteristic is the same as the self intersection number for a compact oriented manifold. However, I'm not sure how to deform the zero set to be transversal. Can anyone get me started please? Thank you! (This question might be related to algebraic geometry, however algebraic geometry is not in our syllabus.)

Edit: the available results from my syllabus are those in Lee ,Hatcher, and Guillemin and Pollack only.

Answer 1

The magic words are "Brieskorn variety". You can look at the original papers to see how people worked with them (Milnor's singular points of complex hypersurfaces is highly recommended, too). See this recent-ish paper: https://arxiv.org/abs/1310.0343