On Gorenstein Fano threefolds with an action of a twodimensional torus

A database of the non-toric, \(\mathbb{Q}\)-factorial, log terminal, Gorenstein, Fano threefolds of Picard number 1 that admit an effective action of a two-dimensional torus. For more information on the provided data see arXiv:2108.03029. This dataset is also available at Github. <strong>Description:</strong> The columns of the csv-file are labelled as follows: <strong>ID:</strong> The ID of the family within the database, numbered 1 through 538. <strong>case:</strong> Refers to the corresponding case in Proposition 2.24. Numbered 1 through 8. <strong>format:</strong> The format of the family as defined on page 13. <strong>generator matrix:</strong> The defining matrix \(P\) as defined in Construction 2.2. <strong>class group: </strong>The isomorphy type of the class group. Presented as \([0, t_1, \dots, t_n]\) meaning \(\mathbb{Z} \oplus ( \mathbb{Z}/t_1 \mathbb{Z} \oplus \dots \oplus \mathbb{Z}/t_n \mathbb{Z} )\). <strong>gd matrix:</strong> Grading matrix of the Cox ring. The \(i\)-th column is the degree of the \(i\)-th Cox ring generator. <strong>relations:</strong> Minimal generating set of the ideal of relations of the Cox ring. <strong>relation degree:</strong> Degree of the minimal generators of the ideal of relations. <strong>anticanonical class:</strong> The anticanonical class. <strong>degree:</strong> Anticanonical self-intersection number \(-\mathcal{K}_X^3\). <strong>gorenstein index:</strong> The smallest positive \(\iota \in \mathbb{Z}\) such that \(\iota \mathcal{K}_X\) is Cartier. Here always equal to 1. <strong>fano index:</strong> The largest \(k \in \mathbb{Z}\) such that \(k D = \mathcal{K}_X\) for some Cartier divisor \(D\). <strong>dimension:</strong> The dimension of the corresponding variety. Here always equal to 3. <strong>picard rank:</strong> The rank of the Picard group. Here always equal to 1. <strong>acr gd matrix:</strong> Grading matrix of the anticanonical ring w.r.t. to a minimal system of generators. The \(i\)-th column is the degree of the \(i\)-th generator. <strong>acr relations:</strong> Minimal generating set of the ideal of relations of the anticanonical ring. <strong>hsNum, hsDenom:</strong> Numerator and denominator of the Hilbert-Poincaré series, i.e. \({\rm HP}_X(t) = {\rm hsNum}(t)/ {\rm hsDenom}(t)\). <strong>hs0 - hs7:</strong> The \(i\)-th coefficient of the power series expansion of \({\rm HP}_X(t)\), i.e. \({\rm HP}_X(t) = {\rm hs0} + {\rm hs1}\, t + {\rm hs2}\, t^2 + \dots + {\rm hs7}\, t^7 + \dots\) <strong>codim:</strong> The codimension of the embedding into a weighted projective space provided by the choice of generators of the anticanonical ring. <strong>genus:</strong> This equals \(h^0 (X, -\mathcal{K}_X)-2\).