Consider a smooth surface of degree . We are interested in determining its cohomology.

**1. A fibration argument**

A key observation is that all such ‘s are diffeomorphic. (When is replaced by , 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 of all degree homogeneous equations, so that is the space of *all* smooth surfaces of degree . There is a universal hypersurface consisting of pairs where is a point lying on the hypersurface . This admits a map

which is (at least intuitively) a fiber bundle over the locus of smooth hypersurfaces. Consequently, if corresponds to smooth hypersurfaces, we get an honest fiber bundle

But is connected, since we have thrown away a *complex* codimension subset to get from ; 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…)