On a special class of smooth codimension two subvarieties in ${\mathbb{P}}^n$, $n\ge 5$

We consider smooth codimension two subcanonical subvarieties in ${\mathbb{P}}^n$ with $n\ge 5$, 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 ${\mathbb{P}}^4$ with small Chern classes.