Arithmetic properties of projective varieties of almost minimal degree

Abstract

ABSTRACT. We study the arithmetic properties of projective varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2. We notably show, that such a variety X ⊂ P r is either arithmetically normal (and arithmetically Gorenstein) or a projection of a variety of minimal degreeX ⊂ P r+1 from an appropriate point p ∈ P r+1 \X. We focus on the latter situation and study X by means of the projectionX → X.If X is not arithmetically Cohen-Macaulay, the homogeneous coordinate ring B of the projecting varietyX is the endomorphism ring of the canonical module K(A) of the homogeneous coordinate ring A of X. If X is non-normal and is maximally Del Pezzo, that is arithmetically Cohen-Macaulay but not arithmetically normal B is just the graded integral closure of A. It turns out, that the geometry of the projectionX → X is governed by the arithmetic depth of X in any case.We study in particular the case in which the projecting varietyX ⊂ P r+1 is a (cone over a) rational normal scroll. In this case X is contained in a variety of minimal degree Y ⊂ P r such that codim Y (X) = 1. We use this to approximate the Betti numbers of X.In addition we present several examples to illustrate our results and we draw some of the links to Fujita's classification of polarized varieties of ∆-genus 1.

On varieties of almost minimal degree in small codimension

“…[2]) derived some information about the Betti numbers of I X for certain varieties X ⊂ P r K of almost minimal degree.…”

Buchsbaum varieties with next to sharp bounds on Castelnuovo-Mumford regularity

“…In this case, the variety V is classified to be a hyperquadric, a (cone over the) Veronese surface, or a rational normal scroll; see [6, (3.10)] and [5, (5.10)]. A nondegenerate projective variety V is a variety of almost minimal degree if deg V = codim V + 2, which is classified to be either a normal Del Pezzo variety or the image of a variety of minimal degree via a projection; see [2,5,16].…”

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

“…Then, by [BS,Theorem 3.1] we know that Sec p (X) = P t−1 K and Σ p (X) ⊆ Sec p (X) is a hyperquadric, where t = depth X p is the arithmetic depth of X p .…”

“…It is an important homological invariant. In the case of a smooth rational normal scrollX we have depth X = dim Σ p (X) + 2 ≤ 4, where Σ p (X) denotes the secant locus ofX with respect to p; see [BS,Theorem 7.5].…”