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On varieties of almost minimal degree in small codimension | | Markus Brodmann
; Peter Schenzel
; | | Date: |
14 Jun 2005 | | Subject: | Commutative Algebra; Algebraic Geometry MSC-class: 14H45; 13D02 | math.AC math.AG | | Affiliation: | Univ. Zürich), Peter Schenzel (Univ. Halle | | Abstract: | The aim of the present exposition is to investigate varieties of almost minimal degree and of low codimension, in particular their Betti diagrams. Here minimal degree is defined as $deg X = codim X + 2.$ We describe the structure of the minimal free resolution of a variety $X$ of almost minimal degree of $codim X leq 4$ by listing the possible Betti diagrams. The most surprising fact is, that the non-arithmetically Cohen-Macaulay case of varieties of almost minimal degree can occur only in small dimensions (cf. Section 2 for the precise statements). Our main technical tool is a result shown by the authors (cf. cite{BS}), which says that besides of an exceptional case, (that is the generic projection of the Veronese surface in $mathbb P^5_K$) any non-arithmetically normal (and in particular non-arithmetically Cohen-Macaulay) variety of almost minimal degree $X subset mathbb P^r_K$ (which is not a cone) is contained in a variety of minimal degree $Y subset mathbb P^r_K$ such that $codim(X,Y) = 1. | | Source: | arXiv, math.AC/0506279 | | Services: | Forum | Review | PDF | Favorites | |
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