a b s t r a c tTo 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. LetX ⊂ P r+1 K be a variety of minimal degree and of codimension at least 2, and consider (2007) [1], it turns out that the cohomological and local properties of X p are governed by the secant locus Σ p (X) ofX with respect to p. \X. By Brodmann and SchenzelAlong these lines, the present paper is devoted to giving a geometric description of the secant stratification ofX , that is of the decomposition of P r+1 K 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 ofX , each of which turns out to be a quasiprojective variety.As an application, we obtain a different geometrical description of non-normal delPezzo varieties X ⊂ P r K , first classified by Fujita (1985) [3, Theorem 2.1(a)] by providing a complete list of pairs (X, p), whereX ⊂ P r+1 K is a variety of minimal degree, p ∈ P r+1 K \X and X p = X ⊂ P r K .

a b s t r a c tLet X ⊂ P r be a variety of almost minimal degree which is the projected image of a rational normal scrollX ⊂ P r+1 from a point p outside ofX . In this paper we study the tangent spaces at singular points of X and the geometry of the embedding scrolls of X , i.e. the rational normal scrolls Y ⊂ P r which contain X as a codimension one subvariety.

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Abstract. Let X ⊂ P r K denote a variety of almost minimal degree other than a normal del Pezzo variety. Then X is the projection of a rational normal scroll\X. We show that the arithmetic depth of X can be expressed in terms of the rank of the matrix M (p), where M is the matrix of linear forms whose 3 × 3 minors define the secant variety ofX.

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