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

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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 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 \(s\le t^2-t\) and \(d\gg t\). In the range \(t^2-2t+3\le s\le t^2-t\), \(d\gg t\), 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.