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 X̃ ⊂ P{doubel-struck}_K^r+1 be a variety of minimal degree and of codimension at least 2, and consider Xp=πp(X̃)⊂P_Kr where p ε P{doubel-struck}_K^r+1\X̃. By Brodmann and Schenzel (2007) [1], it turns out that the cohomological and local properties of Xp are governed by the secant locus σp(X̃) of X̃ with respect to p.Along these lines, the present paper is devoted to giving a geometric description of the secant stratification of X̃, that is of the decomposition of P{doubel-struck}_K^r+1 via the types of secant loci. We show that there are at most six possibilities for the secant locus σ p(X̃), and we precisely describe each stratum of the secant stratification of X̃, 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 ⊂ P{doubel-struck}_Kr, first classified by Fujita (1985) [3, Theorem 2.1(a)] by providing a complete list of pairs (X̃,p), where X̃ ⊂ P{doubel-struck}_K^r+1 is a variety of minimal degree, p ε P{doubel-struck}_K^r+1\X̃ and Xp=X ⊂ P{doubel-struck}_Kr.