Consider a smooth surface {M \subset \mathop{\mathbb P}^3(\mathbb{C})} of degree {d}. We are interested in determining its cohomology.

1. A fibration argument

A key observation is that all such {M}‘s are diffeomorphic. (When {\mathop{\mathbb P}^3} is replaced by {\mathop{\mathbb P}^2}, then this is just the observation that the genus is determined by the degree, in the case of a plane curve.) In fact, consider the space {V} of all degree {d} homogeneous equations, so that {\mathop{\mathbb P}(V)} is the space of all smooth surfaces of degree {d}. There is a universal hypersurface {H \subset \mathop{\mathbb P}^3 \times \mathop{\mathbb P}(V)} consisting of pairs {(p, M)} where {p} is a point lying on the hypersurface {M}. This admits a map

\displaystyle \pi: H \rightarrow \mathop{\mathbb P}(V)

which is (at least intuitively) a fiber bundle over the locus of smooth hypersurfaces. Consequently, if {U \subset \mathop{\mathbb P}(V)} corresponds to smooth hypersurfaces, we get an honest fiber bundle

\displaystyle \pi^{-1}(U) \rightarrow U .

But {U} is connected, since we have thrown away a complex codimension {\geq 1} subset to get {U} from {\mathop{\mathbb P}(V)}; this means that the fibers are all diffeomorphic.

This argument fails when one considers only the real points of a variety, because a codimension one subset of a real variety may disconnect the variety. (more…)