On surfaces of maximal sectional regularity

We study projective surfaces X ⊂ ℙ^r (with r ≥ 5) of maximal sectional regularity and degree d > r, hence surfaces for which the Castelnuovo-Mumford regularity reg(C) of a general hyperplane section curve C = X ∩ ℙ^{r –1} takes the maximally possible value d – r + 3. We use the classi_cation of varieties of maximal sectional regularity of [5] to see that these surfaces are either particular divisors on a smooth rational 3-fold scroll S(1; 1; 1) ⊂ ℙ⁵, or else admit a plane 𝔽 = ℙ² ⊂ ℙ^r such that (Formula Presented) is a pure curve of degree d – r + 3. We show that our surfaces are either cones over curves of maximal regularity, or almost non-singular projections of smooth rational surface scrolls. We use this to show that the Castelnuovo-Mumford regularity of such a surface X satisfies the equality reg(X) = d–r+3 and we compute or estimate various cohomological invariants as well as the Betti numbers of such surfaces.