Projective varieties of maximal sectional regularity

We study projective varieties X⊂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⊂Pⁿ⁺³ or else (b) there is an n-dimensional linear subspace F⊂P^r such that X∩F⊂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.