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 $g\le
Kd^{3/2}thereexistsanirreduciblecomponentoftheHilbertschemeofP^3whosegeneralelementisasmooth,connectedcurveofdegreedandgenusg$ of maximal rank.