ar X iv : m at h / 02 02 09 2 v 1 [ m at h . A G ] 1 1 Fe b 20 02 Fano 3 - folds , K 3 surfaces and graded rings

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 (also known by the older name 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 (you might prefer to call these Q-K3 surfaces); they occur as the general elephant of a Fano 3-fold, but are also interesting in their own right. 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 (and a suitable assumption on the general elephant) 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 determined by orbifold Riemann–Roch (the Lefschetz formula of Atiyah, Singer and Segal, see Reid [YPG]); 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.