The Fano 3-fold database

The Fano 3-fold database

This is a dataset that relates to the graded (homogeneous coordinate) rings of possible algebraic varieties: complex Fano 3-folds with Fano index 1. Each entry in this dataset records the (anticanonical) Hilbert series of a possible Fano 3-fold \(X\), along with the result of some analysis about how \(X\) may be (anticanonically) embedded in weighted projective space \(\mathbb{P}(w_1,w_2,\ldots,w_s)\).

For details, see the paper [BK22], which is a companion and update to the original paper [ABR02].

If you make use of this data, please consider citing [BK22] and the DOI for this data:

doi:10.5281/zenodo.5820338

The data consists of two files in key:value format, "fano3.txt" and "matchmaker.txt". The files "fano3.sql" and "matchmaker.sql" contain the same data as the key:value files, but formatted ready for inserting in sqlite.

fano3.txt

This file contains data that relates to the graded (homogeneous coordinate) rings of possible algebraic varieties. For each entry, the essential characteristic data is the genus and basket; everything else follows (with the exception of the ID). Briefly, this essential data determines a power series, the Hilbert series, \(\text{Hilb}(X,-K_X) = 1 + h_1t + h_2t^2 + \ldots\) that can be written as a rational function of the form \((\text{polynomial numerator in $t$}) / \prod_{i=1}^s(1-t^{w_i})\), where \(w_1,w_2,\ldots,w_s\) are positive integer weights.

The data consists of 52646 entries. The 39550 stable entries (that is, with 'stable' equal to 'true') are assigned an ID 'id' in the range 1-39550. The 13096 unstable entries (that is, with 'stable' equal to 'false') are assigned an ID in the range 41515-54610. IDs in the range 39551-41514 are assigned to the higher index Fano varieties, and are not included in this dataset.

Example entry
id: 1
weights: 5,6,7,...,16
has_elephant: false
genus: -2
h1: 0
h2: 0
...
h10: 4
numerator: t^317 - t^300 - 6*t^299 - ... + 1
codimension: 24
basket: 1/2(1,1,1),1/2(1,1,1),1/3(1,1,2),...,1/5(1,2,3)
basket_size: 7
equation_degrees: 17,18,18,...,27
degree: 1/60
k3_rank: 19
bogomolov: -8/15
kawamata: 1429/60
stable: true

(Some data truncated for readability.)

Brief description of an entry
id: a unique integer ID for this entry
genus: \(h^0(X,-K_X)-2\)
basket: multiset of quotient singularities \(\frac{1}{r}(f,a,-a)\)
basket_size: number of elements in the 'basket'
k3_rank: \(\sum(r-1)\) taken over the 'basket'
kawamata: \(\sum(r-\frac{1}{r})\) taken over the 'basket'
bogomolov: sum of terms over 'basket' relating to stability (see [BK22])
stable: true if and only if 'bogolomov' \(\le0\)
degree: anticanonical degree \((-K_X)^3\) of \(X\), determined by above data (see [BK22])
h1,h2,...,h10: coefficients of \(t,t^2,\ldots,t^{10}\) in the Hilbert series \(\text{Hilb}(X,-K_X)\)
weights: suggestion of weights \(w_1,w_2,\ldots,w_s\) for the anticanonical embedding \(X\subset\mathbb{P}(w_1,w_2,\ldots,w_s)\)
numerator: polynomial such that the Hilbert series \(\text{Hilb}(X,-K_X)\) is given by the power series expansion of \(\text{'numerator'} / \prod_{i=1}^s(1-t^{w_i})\), where the \(w_i\) in the denominator range over the 'weights'
codimension: the codimension of \(X\) in the suggested embedding, equal to \(s - 4\)
has_elephant: true if and only if \(h_1 > 0\)

matchmaker.txt

This file contains a set of pairs of IDs, in each case one from the canonical toric Fano classification [Kas10,toric] and one from "fano3.txt". The meaning is that the Hilbert series of the two agree, and this file contains all such agreeing pairs.

Example entry
toric_id: 1
fano3_id: 27334

Brief description of an entry
toric_id: integer ID in the range 1-674688, corresponding to an 'id' from canonical toric Fano dataset [Kas10,toric]
fano3_id: an integer ID in the range 1-39550 or 41515-54610, corresponding to an 'id' from "fano3.txt"

fano3.sql and matchmaker.sql

The files "fano3.sql" and "matchmaker.sql" contain sqlite-formatted versions of the data described above, and can be imported into an sqlite database via, for example:

$ cat fano3.sql matchmaker.sql | sqlite3 fano3.db

This can then be easily queried. For example:

$ sqlite3 fano3.db
> SELECT id FROM fano3 WHERE degree = 72 AND stable IS TRUE;
39550
> SELECT toric_id FROM fano3totoricf3c WHERE fano3_id = 39550;
547334
547377

References

[ABR02] Selma Altinok, Gavin Brown, and Miles Reid, "Fano 3-folds, K3 surfaces and graded rings", in Topology and geometry: commemorating SISTAG, volume 314 of Contemp. Math., pages 25-53. Amer. Math. Soc., Providence, RI, 2002.
[BK22] Gavin Brown and Alexander Kasprzyk, "Kawamata boundedness for Fano threefolds and the Graded Ring Database", 2022.
[Kas10] Alexander Kasprzyk, "Canonical toric Fano threefolds", Canadian Journal of Mathematics, 62(6), 1293-1309, 2010.
[toric] Alexander Kasprzyk, "The classification of toric canonical Fano 3-folds", Zenodo, doi:10.5281/zenodo.5866330