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

Abstract

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 .

On varieties of almost minimal degree III: Tangent spaces and embedding scrolls

“…So, let depth(X ) = 2 ≤ n. Observe that X is not arithmetically CM so thatX cannot be the Veronese surface in P 5 (see [2,Remark 6.3]). HenceX must be a smooth rational normal scroll.…”

“…. , X 4 ] is generated by two quadrics (see [2,Remark 6.3]); in particular depth(X ) = 3. After an appropriate linear coordinate transformation we may assume that x is the origin of an affine 4-space A 4…”

“…In the ''general case'', namely if the projecting varietyX ⊆ P r+1 is a (cone over) a smooth rational normal scroll this same task turns out to be more demanding. The crucial point here consist in knowing how the so called secant locus Σ p (X) := {q ∈X | #(⟨p, q⟩ ∩X) > 1} ofX with respect to the center of projection p depends on p. In [2] we have solved this problem, by making explicit the so called secant stratification ofX . One application of this is an extension of Fujita's classification of normal del Pezzo varieties to possibly non-normal del Pezzo varieties (see [2]).…”

“…K be either (1) a rational normal curve or (2) a singular curve of almost minimal degree with r ≥ 4.…”

On varieties of almost minimal degree II: A rank-depth formula

“…and (X 0 ) p ⊂ P r K respectively are projections ofX from p and ofX 0 from p 0 , we have (see [BP,Remark 5.4…”