In the study of the geometry of families of algebraic curves one common tool is the comparison between the extrinsic geometry, represented by properties of their projective embeddings and of the Hilbert scheme and the intrinsic geometry, represented by abstract properties and the moduli space. This comparison is often best represented by the natural map π : Hn,g,r →Mg from the Hilbert scheme Hn,g,r of smooth curves in IP r of degree n and genus g (or the Hurwitz scheme for r = 1, the Severi variety for r = 2) to the moduli space of smooth curves of genus g. The behaviour of π is very strongly influenced by the nature of its fibers, the BrillNoether varieties. Now, whileMg is an irreducible variety, the Hilbert scheme Hn,g,r is in general reducible and has in fact many components for r ≥ 3. This difference has inspired many authors in the search of what could be a good component of the Hilbert scheme. One possible answer to such a question is to apply the above principle of comparison ([Ser],