It seems useful to give a counterexample to a commonly misunderstood version of upper semi continuity, namely the one stated erroneously in Shafarevich's BAG chapter I, section 6.3, corollary to theorem 7. the correct version assumes the map is proper or is stated locally on the source instead of the target, as in Mumford's red book. E.g. take source and target both = C^3, complex 3 space, and map (x,y,z)-->(x^2,y,z). The subset W = {(1,y,0)} of the target has reducible preimage, namely two isomorphic copies W1, W2 of W. Now blow up the source along one copy W1, and then remove the exceptional curve over one point of W1. Now the composition of the blow down and the original map, has one dimensional fibers over a non closed subset of the target.

As a remark on the example of the blowup at one point of the second symmetric product of a genus 2 curve over the complexes, this example shows that the second symmetric product of an algebraic curve can have non algebraic deformations, since that is true for the 2 dimensional compact complex torus. This remark has occurred several times since the paper by Xavier Gomez Mont in about 1979.

The lines on a cubic surface are classical, and the conic bundle structure on that surface any one of them gives rise to is also useful. The book by Semple and Roth is laden with classical examples. Perhaps the first occurrence of the use of 4 space to draw conclusions about varieties in 3 space is the paper by Corrado Segre where he studies the quartic surface with a double line in P^3 by projecting to it from an intersection of two quadrics in P^4. This is of course another one in the sequence of del Pezzo surfaces of which the cubic surface is merely the most famous.

The example of Schubert's method of degeneration to count the number of lines in P^3 that meet all of 4 general lines is illustrative. One can also combine it with the Plucker embedding of the set of all lines in P^3, computing the hyperplane section of that embedding, and reduce to the fact that this variety is a quadric in P^5. Joe Harris' book is also loaded with nice examples, including "Fano" parameter spaces and their tangent spaces.