Stratification of the Hilbert scheme of space curves with a stable normal bundle

For d greater or equal to 3g and s between 1 and hg , we study the strata N(d,g,s) of degree d genus g spaces curves C whose normal bundle is stable with stability degree (integer of Lange-Narasimhan) equal to 2s . We prove that N(d,g,s) has an irreducible component of the right dimension whose general curve has a normal bundle with the right number of maximal subbundles. We consider also the semi-stable case (s=0), obtaining similar results. We prove our results by studying the normal bundles of reducible curves and their deformations