A remark on connectedness in hilbert schemes

Let Hd,, denote the Hilbert scheme of locally Cohen-Macaulay curves in P? For any d > 4 and g <_ ( d ; 3 ) , Hd_g has two wll-understood irreducible f an~ i l i tx There is a component E C Hd,, corresponding to extrernal curves (sec [r)]; these are the curves with ~naxirnal Rao function) arid S , the family of sul)r~xtr.en~al curves (see [7]; t h ~ s e have the next largest Rao furiction). 111 this short note we show that S n E # B in Hd., by coristrnctirig an explicit spccialimtiorl (Prop. 1). Our cor~struction also works for AChI curves of gor~us g = ($') + 1 (Remark 2) and lience Hdtg is connected for g > (di3) (Cor0ll;ir). 3).