On the regularities of arithmetically Buchsbaum curves
type: Item
instance of: scholarly article
title: On the regularities of arithmetically Buchsbaum curves
zbMATH Open document ID: 0902.14034
DOI: 10.1007/PL00004331
author: Kohji Yanagawa
published in: Mathematische Zeitschrift
publication date: 1997-12-18T00:00:00Z
review text: Let \(k\) be an algebraically closed field of characteristic zero and \(\mathbb{P}^r\) the projective \(r\)-space over \(k\); let \(X\subset \mathbb{P}^r\) be a closed subscheme. Recall that \(X\) is said arithmetically Buchsbaum if its homogeneous coordinate ring is Buchsbaum. Buchsbaum rings have been intensely studied in recent years, as well as the notion of Castelnuovo-Mumford regularity \(\text{reg} X\) [see for example: \textit{J. Stückrad} and \textit{W. Vogel}, ``Buchsbaum rings and applications. An interaction between algebra, geometry, and topology'' (1986; Zbl 0606.13017) and Math. Ann. 276, 341-352 (1987; Zbl 0628.14037)]. In this paper the author, starting from some results of Stückrad and Vogel (see the above mentioned papers), studies the Castelnuovo-Mumford regularity of an arithmetically Buchsbaum curve; in particular he proves that a nondegenerate integral arithmetically Buchsbaum curve \(C \subset \mathbb{P}^r\) is contained in a surface with minimal degree, when \(\text{reg} C= \lceil (\deg C-1)/(r-1) \rceil +1\) and \(\deg C\gg r\) (where, for a rational number \(x\), one sets \(\lceil x\rceil: =\min \{n\in\mathbb{Z} \mid n\geq x\})\); moreover the author gives a criterion for curves contained in a smooth surface scroll to be arithmetically Buchsbaum.
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 1099268
zbMATH Keywords: Buchsbaum rings
zbMATH Keywords: Castelnuovo-Mumford regularity
zbMATH Keywords: arithmetically Buchsbaum curve
reviewed by: Caterina Cumino
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1099268