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

Abstract

For all integers n, d, g such 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 P n with degree d and 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.