DOI: 10.5281/zenodo.5820338
title: The Fano 3-fold database

description: <strong>The Fano 3-fold database</strong><br> <br> 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. <em><strong>fano3.txt</strong></em> 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. <strong>Example entry</strong><br> id: 1<br> weights: 5,6,7,...,16<br> has_elephant: false<br> genus: -2<br> h1: 0<br> h2: 0<br> ...<br> h10: 4<br> numerator: t^317 - t^300 - 6*t^299 - ... + 1<br> codimension: 24<br> basket: 1/2(1,1,1),1/2(1,1,1),1/3(1,1,2),...,1/5(1,2,3)<br> basket_size: 7<br> equation_degrees: 17,18,18,...,27<br> degree: 1/60<br> k3_rank: 19<br> bogomolov: -8/15<br> kawamata: 1429/60<br> stable: true (Some data truncated for readability.) <strong>Brief description of an entry</strong><br> id: a unique integer ID for this entry<br> genus: \(h^0(X,-K_X)-2\)<br> basket: multiset of quotient singularities \(\frac{1}{r}(f,a,-a)\)<br> basket_size: number of elements in the 'basket'<br> k3_rank: \(\sum(r-1)\) taken over the 'basket'<br> kawamata: \(\sum(r-\frac{1}{r})\) taken over the 'basket'<br> bogomolov: sum of terms over 'basket' relating to stability (see [BK22])<br> stable: true if and only if 'bogolomov' \(\le0\)<br> degree: anticanonical degree \((-K_X)^3\) of \(X\), determined by above data (see [BK22])<br> h1,h2,...,h10: coefficients of \(t,t^2,\ldots,t^{10}\) in the Hilbert series \(\text{Hilb}(X,-K_X)\)<br> weights: suggestion of weights \(w_1,w_2,\ldots,w_s\) for the anticanonical embedding \(X\subset\mathbb{P}(w_1,w_2,\ldots,w_s)\)<br> 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'<br> codimension: the codimension of \(X\) in the suggested embedding, equal to \(s - 4\)<br> has_elephant: true if and only if \(h_1 &gt; 0\) <strong><em>matchmaker.txt</em></strong><br> <br> 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. <strong>Example entry</strong><br> toric_id: 1<br> fano3_id: 27334 <strong>Brief description of an entry</strong><br> toric_id: integer ID in the range 1-674688, corresponding to an 'id' from canonical toric Fano dataset [Kas10,toric]<br> fano3_id: an integer ID in the range 1-39550 or 41515-54610, corresponding to an 'id' from "fano3.txt" <br> <em><strong>fano3.sql </strong></em><strong>and <em>matchmaker.sql</em></strong><br> <br> 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: <pre><code class="language-bash">$ cat fano3.sql matchmaker.sql | sqlite3 fano3.db</code></pre> This can then be easily queried. For example: <pre><code class="language-bash">$ sqlite3 fano3.db &gt; SELECT id FROM fano3 WHERE degree = 72 AND stable IS TRUE; 39550 &gt; SELECT toric_id FROM fano3totoricf3c WHERE fano3_id = 39550; 547334 547377</code></pre> <strong>References</strong> [ABR02] Selma Altinok, Gavin Brown, and Miles Reid, "Fano 3-folds, K3 surfaces and graded rings", in <em>Topology and geometry: commemorating SISTAG</em>, volume 314 of <em>Contemp. Math.</em>, pages 25-53. Amer. Math. Soc., Providence, RI, 2002.<br> [BK22] Gavin Brown and Alexander Kasprzyk, "Kawamata boundedness for Fano threefolds and the Graded Ring Database", 2022.<br> [Kas10] Alexander Kasprzyk, "Canonical toric Fano threefolds", <em>Canadian Journal of Mathematics</em>, 62(6), 1293-1309, 2010.<br> [toric] Alexander Kasprzyk, "The classification of toric canonical Fano 3-folds", <em>Zenodo</em>, doi:10.5281/zenodo.5866330