On a special class of smooth codimension two subvarieties in Pn, n⩾5

We consider smooth codimension two subcanonical subvarieties in Pn with n⩾5, lying on a hypersurface of degree s having a linear subspace of multiplicity s-2. We prove that such varieties are complete intersections. We also give a little improvement to some earlier results on the non existence of rank two vector bundles on P4 with small Chern classes.