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

Abstract 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 ⊂ P K r + 1 be a variety of minimal degree and of codimension at least 2, and consider X p = π p ( X ) ⊂ P K r where p ∈ P K r + 1 ∖ X . By Brodmann and Schenzel (2007) [1] , it turns out that the cohomological and local properties of X p are governed by the secant locus Σ p ( X ) of X with respect to p . Along these lines, the present paper is devoted to giving a geometric description of the secant stratification of X , 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 Σ p ( X ) , and we precisely describe each stratum of the secant stratification of X , 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 K r , first classified by Fujita (1985) [3, Theorem 2.1(a)] by providing a complete list of pairs ( X , p ) , where X ⊂ P K r + 1 is a variety of minimal degree, p ∈ P K r + 1 ∖ X and X p = X ⊂ P K r .