Projective varieties of maximal sectional regularity

We study projective varieties X subset of P-r of dimension n >= 2, of codimension c >= 3 and of degree d >= c+3 that are of maximal sectional regularity, i.e. varieties for which the Castelnuovo-Mumford regularity reg(C) of a general linear curve section is equal to d-c+1, the maximal possible value (see [10]). As one of the main results we classify all varieties of maximal sectional regularity. If X is a variety of maximal sectional regularity, then either (a) it is a divisor on a rational normal (n + 1)-fold scroll Y subset of Pn+3 or else (b) there is an n-dimensional linear subspace F subset of P-r such that X boolean AND F subset of F is a hypersurface of degree d-c+1. Moreover, suppose that n = 2 or the characteristic of the ground field is zero. Then in case (b) we obtain a precise description of X as a birational linear projection of a rational normal n-fold scroll. (C) 2016 Elsevier B.V. All rights reserved.