On Surfaces of Maximal Sectional Regularity

Abstract

We study projective surfaces X subset of 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 boolean AND Pr-1 takes the maximally possible value d-r + 3. We use the classification 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) subset of P-5, or else admit a plane F = P-2 subset of P-r such that X boolean AND F subset of F 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.