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29 March 2024
 
  » arxiv » math.AC/0506277

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Arithmetic properties of projective varieties of almost minimal degree
Markus Brodmann ; Peter Schenzel ;
Date 14 Jun 2005
Subject Commutative Algebra; Algebraic Geometry MSC-class: 14H45; 13D02 | math.AC math.AG
AffiliationUniv. Zürich), Peter Schenzel (Univ. Halle
AbstractWe 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 $X subset {mathbb P}^r$ is either arithmetically normal (and arithmetically Gorenstein) or a projection of a variety of minimal degree $ ilde {X} subset {mathbb P}^{r + 1}$ from an appropriate point $p in {mathbb P}^{r + 1} setminus ilde {X}$. We focus on the latter situation and study $X$ by means of the projection $ ilde {X} o X$. If $X$ is not arithmetically Cohen-Macaulay, the homogeneous coordinate ring $B$ of the projecting variety $ ilde {X}$ 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 projection $ ilde {X} o X$ is governed by the arithmetic depth of $X$ in any case. We study in particular the case in which the projecting variety $ ilde {X} subset {mathbb P}^{r + 1}$ is a cone (over a) rational normal scroll. In this case $X$ is contained in a variety of minimal degree $Y subset {mathbb P}^r$ 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 $Delta $-genus 1.
Source arXiv, math.AC/0506277
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