In textbooks where sheaf cohomology is introduced, example computations are typically carried out on known spaces, e.g. $\mathbb{P}^n$.
However, suppose we are handed a description of a variety solely by its defining equation, e.g.,
$$v^2w + w^2x + x^2y + y^2z + z^2v = 0$$
where $x,y,z,w,v$ are homogeneous coordinates on $\mathbb{P}^4$. How does one then go about finding different sheaf cohomology groups or for example the Hodge diamond, when what the space “is” is not explicit?(The example is the Klein cubic 3-fold.)
My initial guess is to see if it’s reducible, etc, and try to find an acyclic covering, from which I could use the Cech complex.
(For example for $x^2 + y^2 + z^2=0$ in $\mathbb{A}^3_{\mathbb{R}}$, we would use $X = S^2$.)