Projective surfaces of degree r+1 in projective r-space and almost non-singular projections

We study projective surfaces of degree r+1 in projective r-space, more precisely (non-conic) irreducible non-degenerate surfaces X⊂Pr of degree r+1. We may divide up the class of these surfaces in surfaces whose affine cone satisfies the second Serre property S2 and surfaces which occur as almost non-singular projections of either a smooth rational scroll or else of a del Pezzo surfaces which is arithmetically Cohen–Macaulay. We focus on those surfaces which occur as almost non-singular projections and study their geometric and cohomological properties.