Graded rings and birational geometry

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.