Definition: A smooth curve C in P³ is said to be projectively normal if h¹(I_C(m))=0 for every m in Z.

Observe that a projectively normal (p.n.) curve is connected (h¹(I_C) =0).

If C is p.n. its minimal free resolution has length one, i.e. is of the form:

Indeed, assume that 0 -> E -> +O(-n_1i) -> I_C-> 0 is the beginnning of the m.f.r., then E is locally free (i.e. E is a vector bundle). Since this is the m.f.r., the map H⁰(+O(m-n_1i)) -> H⁰(I_C(m)) is surjective for every m, so h¹(E(m))=0 for every m. Moreover, since C is p.n., h²(E(m))=0, for every m. From Horrock's theorem (see for example the book of Okonek-Schneider-Spindler: "Vector bundles on complex projective spaces", Prog. in Math, 3, Birkhaeuser) it follows that E is totally split (E = +O(l_i)).

Conversely it is clear that if the m.f.r. of C has length one, then C is p.n. In conclusion, a smooth curve C in P³ is projectively normal iff its m.f.r. has length one.

Another characterization goes as follows: consider the exact sequence of restriction to a plane H:

Conversely if H⁰(I_C(m)) -> H⁰(I_CH(m)) is surjective for every m, since h¹(I_C(m))=0 for m >> 0, one easily deduces that C is p.n.

It follows that, if C is p.n., the m.f.r. of CH is the restriction to H of the m.f.r. of C (so to get the m.f.r. of C it is enough to have the m.f.r. of the set of points CH).

What has been said so far can be extended to singular (even non-reduced) curves. A curve C in P³ is said to be arithmetically Cohen-Macaulay (a.C.M.) if it is locally Cohen-Macaulay (i.e. all the local rings O_C,x are Cohen-Macaulay; in this case (C is of dimension one) this means that C has no embedded points, in particular every reduced curve is loc.C.M.), and if h¹(I_C(m))=0 for every m in Z. Every consideration made above can be repeated in this broader context: a p.n. curve in P³ is an a.C.M. curve which is smooth. For example every complete intersection curve is a.C.M.