Non-degenerate curves with maximal Hartshorne-Rao module

Extending results for space curves we establish bounds for the cohomology of a non-degenerate curve in projective n-space. As a consequence, for any given n we determine all possible pairs (d, g) where d is the degree and g is the (arithmetic) genus of the curve. Furthermore, we show that curves attaining our bounds always exist and describe properties of these extremal curves. In particular, we determine the Hartshorne-Rao module, the generic initial ideal and the graded Betti numbers of an extremal curve.