Projective varieties of maximal sectional regularity

We study projective varieties X⊂PrX⊂Pr of dimension n≥2n≥2, of codimension c≥3c≥3 and of degree d≥c+3d≥c+3 that are of maximal sectional regularity, i.e. varieties for which the Castelnuovo–Mumford regularity reg(C)reg(C) of a general linear curve section is equal to d−c+1d−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)(n+1)-fold scroll Y⊂Pn+3Y⊂Pn+3 or else (b) there is an n  -dimensional linear subspace F⊂PrF⊂Pr such that X∩F⊂FX∩F⊂F is a hypersurface of degree d−c+1d−c+1. Moreover, suppose that n=2n=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.