This database contains lists which should be interpreted in different ways. This page explains the different types of lists that occur and gives warnings about the data in particular databases.

Different types of list

Some of the lists in this database are complete classifications. Others may be sublists of complete classifications or overlists, in which every member of the classification occurs but other unwanted members may be listed too. There are also various combinations of these.

• Classification: The list describes every member of a mathematically-specified class of objects, and every member on the list is realised by an object of the class. Classifications include:
  • Toric terminal Fano 3-folds;
  • Toric canonical Fano 3-folds.
• Part of a filtered classification: Initial parts of classifications include:
• Toric log del Pezzo surfaces; this is filtered by a height function and includes those surfaces of heights up to 17.
• Smooth toric Fano varieties; this is filtered by dimension and includes those varieties of dimension up to 5.
• .
• Fano 4-fold terminal wps.
• Sublist
• Canonical 3-folds.
• Fano 4-fold quasismooth hypersurfaces.
• Overlist
• Fano 3-folds.
• Hybrids
• K3 surfaces.

Warnings about individual databases

K3 database

This is a hybrid of sublist and overlist. There is no guarantee that a Hilbert series listed in the K3 database is that of a polarised K3 surface X,A. However, in principle every case could be checked, and some cases are fairly easy:

• If the model given is in codimension at most 4, then there is a K3 surface in that model realising the Hilbert series.
• Torelli for K3 surfaces is easy to apply for the general member of any family with fairly small singular rank. In that case, it is usually easy to see that there a K3 surface exists with the correct polarisation and basket, and so also with the right Hilbert series. However, this method does not confirm that there is a K3 surface in the given model.

Conversely, since there are countably many families of polarised K3 surfaces but the database is a finite list of Hilbert series, not every polarised K3 surface appears in the list. However, the list is an overlist of all polarised K3 surfaces under the following interpretation: if X,A has genus g = h⁰(X,A)-1, then

• if g = -1,0,1 or 2, then the Hilbert series of X,A appears in the list.
• if g > 2, then the degree and Hilbert series differs from the g = 2 case only by A^2 = d + 2(g-2) and P(X,A)(t) = Q(t) + (g-2)t(1+t)/(1-t)^3, where d and Q(t) are the degree and Hilbert series computed using the same basket as X,A but with genus set to 2. In this sense, the degree and Hilbert series of any K3 surface is represented on the table.
• A number of cases with g > 2 appear in the lists in any case: those with a model in low codimension have been included so that all families in low codimension are listed. This is most clearly seen in the big table.

Fano 3-folds

The number of number deformation families of Fano 3-folds is known to be finite; this is originally by Kawamata. For Fano 3-folds of index f = 1 (that is, those for which -K_X is not divisible as a Weil divisor class) the list is simply an overlist of the true classification. Thus

1. If X is a Fano 3-fold of index 1, then its Hilbert series will be in the database.
2. The converse is not proved and almost certainly not true: that is, a Hilbert series on the database cannot be assumed to be that of a Fano 3-fold, although the following have been checked:
   • Those Hilbert series with candidates in codimensions 1-4 do exist with the suggested models
   • Around 6,000 of the Fano Hilbert series are those of toric Fano 3-folds (but with high index and usually with canonical singularities); these are indicated in the additional output for each Fano 3-fold.

When the index is higher, f > 2, then the situation is similar but complicated by two factors:

1. Missing: Fano 3-folds of index 2 that are not stable (in the sense of Kawamata) are not included in the list (although the unstable ones in higher index are).
2. Bad weights: The weights assigned to candidates are done more naively than when f = 1 and so are not a useful guide in codimension 4 or higher. This is only a fault of the candidates and not the Hilbert series: these Fano 3-folds may well exist, but probably not in the model suggested. (Improbable candidates are usually clear: embeddings in P⁷(1²,2²,3²,18,19) are absurd: if this Fano exists, one would expect additional generators in degrees 4..17. A comparison with expected embeddings for a K3 elephant are carried out in [BS07a].)