Important warnings

The graded ring method

The lists in these pages are based on power series generated by Riemann--Roch formulas, an approach sometimes called the graded ring method. Such a power series may be the Hilbert series of a polarised variety X,A, and the challenge then is to exhibit the homogeneous coordinate ring of X,A. From this point of view, the graded ring method comes in two parts. The first is to understand RR and the combinatorics of its input, and this is what is worked out on these pages (based on published RR formulas). The second part is to construct a graded ring. This involves the commutative algebra of Gorenstein rings, although in particular cases there may be other techniques. This second part is not worked out by the routines here.

Results are usually presented by proposing a description of a polarised variety X,A, known as a candidate, that should have the right Hilbert series. In general, one should not assume that any variety with the right Hilbert series exists, or, if it does exist, that it resembles the candidate. Indeed, it is certain that some of the more complicated candidates do not actually exist. This is most important to bear in mind when looking at lists of candidates for Fano 3-folds.

Fortunately, there are many cases where the existence of a polarised variety X,A with the given Hilbert series is known and agrees with the candidate. In codimensions 1, 2 and 3, one can either write down explicit graded rings with the desired properties or see that no such ring is possible (in the form the candidate implies). For K3 surfaces (and also some Fanos) in codimension 4, Altinok checks the existence. In codimension 5, Frantzen makes more calculations for K3 surfaces while Buckley computes some Calabi-Yau 3-folds.

In short, the warning so far is that results are really Hilbert series, not varieties, and there is no claim that a corresponding Hilbert scheme has a component containing quasi-smooth varieties. Equally, when a variety does exist, there is no claim that the family of all such varieties is irreducible. This holds for K3 surfaces by the Torelli theorem, but is known to be false for Fano 3-folds by examples of Takagi.

Issues for lists of particular types of variety are described below.

K3 surfaces

The lists up to codimension 4 have been checked. Candidates in higher codimension are generated by assuming that any unprojection of Type I or Type II₁ works as expected whenever the numerical properties of the basket and ambient weights make it a possibility. The Torelli theorem is not used in a systematic way.

Fano 3-folds

The current list is faulty: it has 93 rather than 95 Fanos in codimension 1, for instance. And presumably many of its more complicated candidates do not exist as Fano 3-folds.

Calabi-Yau 3-folds

The examples here are a preliminary database based on examples in Buckley's PhD thesis. It contains only 48 candidates found by making minor modifications to Buckley's data.

This small initial list of candidates is mainly given as motivation for us to find some interesting examples. We are well aware of the large lists of Calabi-Yau 3-folds, and of course we do not intend to duplicate them (nor are we able to). We would be grateful for more examples of the graded rings of Calabi-Yau 3-folds in low codimension, especially if their equations can be written in novel formats.