On varieties of almost minimal degree I: Secant loci of rational normal scrolls

To provide a geometrical description of the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let (X) over bar subset of P-K(r+1) be a variety of minimal degree and of codimension at least 2, and consider X-p = pi(p)((X) over bar) subset of P-K(r) where p is an element of P-K(r+1) \ (X) over bar. By Brodmann and Schenzel (2007) [1], it turns out that the cohomological and local properties of X-p are governed by the secant locus Sigma(p)((X) over bar) of (X) over bar with respect to p. Along these lines, the present paper is devoted to giving a geometric description of the secant stratification of (X) over bar, that is of the decomposition of P-K(r+1) via the types of secant loci. We show that there are at most six possibilities for the secant locus Sigma(p)((X) over bar), and we precisely describe each stratum of the secant stratification of (X) over bar, each of which turns out to be a quasi-projective variety. As an application, we obtain a different geometrical description of non-normal del Pezzo varieties X subset of P-K(r). first classified by Fujita (1985) [3, Theorem 2.1(a)] by providing a complete list of pairs ((X) over bar, p), where (X) over bar subset of P-K(r+1) is a variety of minimal degree, p is an element of P-K(r+1) \ (X) over bar and X-p = X subset of P-K(r). (C) 2010 Elsevier B.V. All rights reserved