Good components of curves in projective spaces outside the Brill-Noether range

For all integers n,d,gsuch that n≥4, d≥n+ 1, and (n+ 2)(d−n−1) ≥n(g−1), we define a good (i.e. generically smooth of dimension (n+ 1)d+ (3 −n)(g−1) and with the expected number of moduli) irreducible component A(d,g; n) of the Hilbert scheme of smooth and nondegenerate curves in Pnwith degree dand genus g. For most of these (d,g), we prove that a general X∈A(d,g; n) has maximal rank. We cover, in this way, a range of (d,g,n) outside the Brill–Noether range.