On the projective genus of surfaces

Let X ⊂ PN be a smooth irreducible non degenerate surface over the complex numbers, N ≥ 4. We define the projective genus of X, denoted by PG(X), as the geometric genus of the singular curve of the projection of X from a general linear subspace of codimension four. Denote by g(X) the sectional genus of X. In this paper we conjecture that the only surfaces for which PG(X) = g(X) -1 are the del Pezzo surface in ℙ4, in ℙ5 and a conic bundle of degree 5inP4. We prove that for N ≥ 5 if PG(X) = g(X) -1 + λ, λ a non negative integer, then g(X) ≤ λ + 1 + a where a = -2 for a scroll and a = 0 otherwise, and deduce the conjecture for N ≥ 5 from this statement.