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 be a variety of minimal degree and of codimension at least 2, and consider where . By Brodmann and Schenzel (2007) [1], it turns out that the cohomological and local properties of Xp are governed by the secant locus of with respect to p.Along these lines, the present paper is devoted to giving a geometric description of the secant stratification of , that is of the decomposition of via the types of secant loci. We show that there are at most six possibilities for the secant locus , and we precisely describe each stratum of the secant stratification of , 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 , first classified by Fujita (1985) [3, Theorem 2.1(a)] by providing a complete list of pairs , where is a variety of minimal degree, and .