  In this article we study, given a pair of integers (d,g), the problem of
existence of a smooth, irreducible, non-degenerate curve in the projective
n-domensional space of degree d and genus g (the Halphen-Castelnuovo Problem).
We define two domains from the (d,g)-plane, D1,n and D2,n, and we prove that
there is no gap in D1,n. This follows by constructing curves on some rational
surfaces with hyperelliptic hyperplane sections, and from some previous
Theorems of Ciliberto, Sernesi, and of the author. Moreover, in the last
section, based on some results of Horrowitz, Ciliberto, Harris, Eisenbud, we
Conjecture that D2,n is the right lacunary domain.
