Dataset Open Access

Brown, Gavin; Kasprzyk, Alexander M.

**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

Files
(142.5 MB)

Name | Size | |
---|---|---|

fano3.sql
md5:bced9fca984dd1cee5d6d4a48102a416 |
49.9 MB | Download |

fano3.txt
md5:fee6b5fd94c32bc10da7b7f12616e4de |
59.1 MB | Download |

LICENSE
md5:65d3616852dbf7b1a6d4b53b00626032 |
7.0 kB | Download |

matchmaker.sql
md5:4df6858e99c8f5e642fb3bc193d5147e |
10.7 MB | Download |

matchmaker.txt
md5:b8408481f64d7e8509e33c7ebc991544 |
22.8 MB | Download |

README.txt
md5:61c254abd2ad581e86c674381beb359b |
4.9 kB | Download |

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