  Let $C$ be an irreducible, reduced, non-degenerate curve, of arithmetic genus
$g$ and degree $d$, in the projective space $\mathbf P^4$ over the complex
field. Assume that $C$ satisfies the following {\it flag condition of type
$(s,t)$}: {$C$ does not lie on any surface of degree $<s$, and on any
hypersurface of degree $<t$}. Improving previous results, in the present paper
we exhibit a Castelnuovo-Halphen type bound for $g$, under the assumption
$s\leq t^2-t$ and $d\gg t$. In the range $t^2-2t+3\leq s\leq t^2-t$, $d\gg t$,
we are able to give some information on the extremal curves. They are
arithmetically Cohen-Macaulay curves, and lie on a flag like $S\subset F$,
where $S$ is a surface of degree $s$, $F$ a hypersurface of degree $t$, $S$ is
unique, and its general hyperplane section is a space extremal curve, not
contained in any surface of degree $<t$. In the case $d\equiv 0$ (modulo $s$),
they are exactly the complete intersections of a surface $S$ as above, with a
hypersurface. As a consequence of previous results, we get a bound for the
speciality index of a curve satisfying a flag condition.

Algebraic Geometry