Fano 3-folds, K3 surfaces and graded rings

By S. Altınok, Gavin Brown, M. Reid. 2002

90 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f483c9e45f27874f0ec90771a13d1463c2e06848

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Three-fold divisorial contractions to singularities of higher indices (2003, background): “We have positive answers [27], [29] and [31] also for Fano three-folds with mild singularities, whereas there exist counter-examples in [2] when the anticanonical system has small dimension.”
Birational Geometry of 3-fold Mori Fibre Spaces (2003, methodology): “Starting from Y 2,3, we construct Y as an unprojection, that is, the contraction of the Weil divisor � ⊂ Y ; see [Rei], [ ABR02 ] for details.”