A remark on the genus of curves in $\mathbf P^4$

Abstract

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.