Fano 3-folds, K3 surfaces and graded rings

Brown, Gavin D. and Altmok, Selma and Reid, Miles (2002) Fano 3-folds, K3 surfaces and graded rings. Contemporary Mathematics, 314 . pp. 25-61. ISSN 0271-4132. (Full text available)

PDF (Fano 3-folds)
Download (411kB) Preview


Explicit birational geometry of 3-folds represents a second phase of Mori theory, going beyond the foundational work of the 1980s. This paper is a tutorial and colloquial introduction to the explicit classification of Fano 3-folds (Q-Fano 3-folds), a subject that we hope is nearing completion. With the intention of remaining accessible to beginners in algebraic geometry, we include examples of elementary calculations of graded rings over curves and K3 surfaces. For us, K3 surfaces have at worst Du Val singularities and are polarised by an ample Weil divisor; they occur as the general elephant of a Fano 3-fold. A second section of the paper runs briefly through the classical theory of nonsingular Fano 3-folds and Mukai's extension to indecomposable Gorenstein Fano 3-folds. Ideas sketched out by Takagi at the Singapore conference reduce the study of Q-Fano 3-folds with g>=2 to indecomposable Gorenstein Fano 3-folds together with unprojection data. Much of the information about the anticanonical ring of a Fano 3-fold or K3 surface is contained in its Hilbert series. The Hilbert function is given by orbifold Riemann--Roch (see Reid's Young Person's Guide); using this, we can treat the Hilbert series as a simple collation of the genus and a basket of cyclic quotient singularities. Many hundreds of families of K3s and Fano 3-folds are known, among them a large number with g<=0, and Takagi's methods do not apply to these. However, in many cases, the Hilbert series already gives firm indications of how to construct the variety by biregular or birational methods. A final section of the paper introduces the K3 database in Magma, that manipulates these huge lists without effort.

Item Type: Article
Additional information: Proceedings of SISTAG, Singapore 2001
Uncontrolled keywords: Algebraic Geometry (math.AG)
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Judith Broom
Date Deposited: 19 Dec 2007 18:26 UTC
Last Modified: 12 May 2014 15:23 UTC
Resource URI: (The current URI for this page, for reference purposes)
  • Depositors only (login required):


Downloads per month over past year