Maximal rank of space curves in the range A

Abstract

We prove the following statement, which has been conjectured since 1985: There exists a constant $K$ such that for all natural numbers $d,g$ with $g\le K{d}^{3/2}$ there exists an irreducible component of the Hilbert scheme of ${\mathbb{P}}^3$ whose general element is a smooth, connected curve of degree $d$ and genus $g$ of maximal rank.