Characterizations of space curves containing a planar subcurve

We introduce and study the space curves of “$h$-extremal type”, that are curves of degree $d$ and arithmetic genus $g$ whose Rao function agrees, in a suitable interval depending on $d$ and $h$, with the one of the “$h$-extremal” curves introduced by NotariSabadini.
Our study is motivated by the literature of the last years concerning curves with large cohomology and their relations with the Hilbert scheme.
Our main result is a Structure Theorem which gives some geometrical characterizations of such curves. The most intriguing is that if $d$ is sufficiently large with respect to $h$, a curve of hextremal type contains a planar subcurve of degree $d$ --- $h$ h and lies on a non integral quadric. As a consequence we can determine all possible Rao functions (for fixed $d, g, h$). We add several examples which show, in particular, that our result is the best possible for $h \leq$ 5.