Towards a general theory of unprojection
type: Item
instance of: scholarly article
title: Towards a general theory of unprojection
zbMATH Open document ID: 1145.14039
author: Stavros Argyrios Papadakis
published in: Journal of Mathematics of Kyoto University
publication date: 2008-07-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0608158
review text: Let \(R\) be a Gorenstein ring. \textit{Unprojection} is a method to construct a new Gorenstein ring \(S\) of codimension bigger than the codimension of \(R\). From the algebraic point of view it is a substitute for a structure theory of Gorenstein rings in low codimension. From the geometric point of view it gives a graded ring method to construct terminal \(\mathbb{Q}\)-factorial Fano 3-folds and Sarkisov birational links between them. There are currently four types of unprojection, namely unprojection of types \textit{I, II, III} and \textit{IV}. The simplest case is unprojection of type \textit{I} due \textit{A. Kustin} and \textit{M. Miller} [J. Algebra 85, 303--322 (1983; Zbl 0522.13011)] and \textit{S. A. Papadakis} and \textit{M. Reid} [J. Algebr. Geom. 13, No. 3, 563--577 (2004; Zbl 1071.14047)]. In previous works the author has also studied cases of unprojection of types \textit{II} and \textit{III}. In this paper the author introduces a general definition of unprojection. In particular, given a commutative ring with unit \(\mathcal{O}_X\), and \(I_D \subset \mathcal{O}_X\) an ideal containing an \(\mathcal{O}_X\)-regular element \(q \in I_D\), he defines the unprojection algebra \(\mathrm{unpr}_{\mathcal{O}_X}I_D\) and he shows that this definition agrees with the definitions given in the special cases treated in previous works by himself and M. Reid.
reviewed by: Nikolaos Tziolas
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 5302028
zbMATH Keywords: unprojection
zbMATH Keywords: Fano 3-folds
zbMATH Keywords: Gorenstein rings
describes a project that uses: SINGULAR
MaRDI profile type: Publication
OpenAlex ID: W2159941523
arXiv ID: math/0608158
DOI: 10.1215/KJM/1250281025
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description: scientific article