I'm learning about Lefschetz pencils and vanishing cycles and have looked at a few sources:

- http://www.math.purdue.edu/~dvb/preprints/sheaves.pdf
- http://www3.nd.edu/~lnicolae/Morse2nd.pdf
- Voisin's Complex Algebraic Geometry and Hodge Theory II

I want to be able to start computing explicit examples of vanishing cycles and monodromy, but I need to first learn how to compute the singular homology of complex varieties. What are some tools I can use to compute cohomology of some smooth projective varieties over $\mathbb{C}$? For example, how can I study the cohomology of some Lefschetz pencil for the projective scheme $$ \begin{matrix} \textbf{Proj}\left( \frac{\mathbb{C}[x,y,z,w]}{(x^8 + y^8 + z^8 + w^8 +x^2y^2z^2w^2)} \right) \end{matrix} $$