On Buchsbaum rings obtained by gluing

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.

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“…We say that x ∈ X is a Goto or G-point, if the local ring O X,x is of "Goto type" (cf [17]) thus if dim O X,x > 1 and …”

Section: But This Latter Map Is An Isomorphism If and Only Ifmentioning

confidence: 99%

Arithmetic properties of projective varieties of almost minimal degree

Brodmann

Schenzel

2006

J. Algebraic Geom.

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“…We then have H i m (A) = (0) for all i ∈ {1, d} and H 1 m (A) = S/A, because depth A S = d and ℓ A (S/A) < ∞. Hence, depth A = 1, and A is a Buchsbaum ring ( [4,5]). We have S = m : m, and Ass A = Ass A S = Assh A, since depth A S = d. Thus, we get the following.…”

Section: 1mentioning

confidence: 99%