Resumen de Characterizations of spaces curves containing a planar subcurve

Silvio Greco , Giuseppe Paxia

• 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 Notari-Sabadini. 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 h-extremal type contains a planar subcurve of degree d - h and lies on a non integral quadric. As a consequence we can determine all possible Rao functions (for fxed d, g, h). We add several examples which show, in particular, that our result is the best possible for h = 5.