Reid, Miles (2000) Graded rings and birational geometry. In: Algebraic Geometry Symposium, Kinosaki, Japan, Oct 2000. Published in: Proceedings of Algebraic Geometry Symposium pp. 1-72.

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Abstract

This paper is a written version of my lecture \Rings and varieties" at the Kinosaki algebraic geometry workshop in Oct 2000, and a series of two lectures at Tokyo University in Dec 2000. It is intended to be informative and attractive, rather than strictly accurate, and I expect it to stimulate work in a rapidly developing eld (as did its predecessor Reid [R3]). The paper was prepared in a hurry to meet a deadline, and one or two sections remain in rst draft. I apologise to the reader and the referee for any inconvenience caused. The canonical ring of a regular algebraic surface of general type, the graded ring over a K3 surface with Du Val singularities polarised by an ample Weil divisor, or the anticanonical ring of a Fano variety is a Gorenstein ring. In simple cases, a Gorenstein ring is a hypersurface, a codimension 2 complete intersection, or a codimension 3 Pfa an. We now have additional techniques based on the idea of projection in birational geometry that produce results in codimension 4 (and 5, etc.), even though there is at present no useable structure theory for the graded ring. This paper applies graded ring methods, especially unprojection, to the existence of Fano 3-folds and of Sarkisov birational links between them. The 3-fold technology applies also to some extent to construct canonical surfaces. A recurring theme is that unprojection often acts as a working substitute for a structure theory of Gorenstein rings in low codimension. I discuss what little I understand of the structure of codimension 4 Gorenstein rings, and present a general and entirely useless structure theorem. The nal section of the paper contains a brief outline of forthcoming joint work with Gavin Brown on C covers of Mori ips of Type A, intended to illustrate the use of serial unprojection.

Item Type:	Conference Item (Paper)
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Graded rings, Geometry, Algebraic , Threefolds (Algebraic geometry), Surfaces, Algebraic
Series Name:	Departmental Bulletin Paper
Journal or Publication Title:	Proceedings of Algebraic Geometry Symposium
Editor:	Ohno, K.
Official Date:	2000
Dates:	Date Event 2000 Published
Page Range:	pp. 1-72
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)
Description:	Japanese preprint series
Date of first compliant deposit:	17 December 2021
Date of first compliant Open Access:	20 December 2021
Conference Paper Type:	Paper
Title of Event:	Algebraic Geometry Symposium
Type of Event:	Other
Location of Event:	Kinosaki, Japan
Date(s) of Event:	Oct 2000
Related URLs:	Other Repository

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