On Gorenstein Fano threefolds with an action of a twodimensional torus
2022-11-07

&lt;p&gt;A database of the non-toric, &lt;span class="math-tex"&gt; &lt;/span&gt;-factorial, log terminal, Gorenstein, Fano threefolds of Picard number 1 that admit an effective action of a two-dimensional torus.&lt;/p&gt; &lt;p&gt;For more information on the provided data see &lt;a href="https://arxiv.org/abs/2108.03029"&gt;arXiv:2108.03029&lt;/a&gt;. This dataset is also available at &lt;a href="https://github.com/abaeuerle/fano-3d-lt-gor-rho-1"&gt;Github&lt;/a&gt;.&lt;/p&gt; &lt;p&gt;&lt;strong&gt;Description:&lt;/strong&gt;&lt;/p&gt; &lt;p&gt;The columns of the csv-file are labelled as follows:&lt;/p&gt; &lt;ul&gt; &lt;li&gt;&lt;strong&gt;ID:&lt;/strong&gt;&nbsp;The ID of the family within the database, numbered 1 through 538.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;case:&lt;/strong&gt;&nbsp;Refers to the corresponding case in Proposition 2.24. Numbered 1 through 8.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;format:&lt;/strong&gt; The format of the family as defined on page 13.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;generator matrix:&lt;/strong&gt;&nbsp;The defining matrix&nbsp;&lt;span class="math-tex"&gt; &lt;/span&gt;&nbsp;as defined in Construction 2.2.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;class group:&nbsp;&lt;/strong&gt;The isomorphy type of the class group. Presented as&nbsp;&lt;span class="math-tex"&gt; &lt;/span&gt;&nbsp;meaning&nbsp;&lt;span class="math-tex"&gt; &lt;/span&gt;.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;gd matrix:&lt;/strong&gt;&nbsp;Grading matrix of the Cox ring. The &lt;span class="math-tex"&gt; &lt;/span&gt;-th column is the degree of the &lt;span class="math-tex"&gt; &lt;/span&gt;-th Cox ring generator.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;relations:&lt;/strong&gt;&nbsp;Minimal generating set of the ideal of relations of the Cox ring.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;relation degree:&lt;/strong&gt;&nbsp;Degree of the minimal generators of the ideal of relations.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;anticanonical class:&lt;/strong&gt;&nbsp;The anticanonical class.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;degree:&lt;/strong&gt;&nbsp;Anticanonical self-intersection number&nbsp;&lt;span class="math-tex"&gt; &lt;/span&gt;.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;gorenstein index:&lt;/strong&gt;&nbsp;The smallest positive&nbsp;&lt;span class="math-tex"&gt; &lt;/span&gt;&nbsp;such that&nbsp;&lt;span class="math-tex"&gt; &lt;/span&gt;&nbsp;is Cartier. Here always equal to&nbsp;1.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;fano index:&lt;/strong&gt;&nbsp;The largest&nbsp;&lt;span class="math-tex"&gt; &lt;/span&gt;&nbsp;such that&nbsp;&lt;span class="math-tex"&gt; &lt;/span&gt; for some Cartier divisor &lt;span class="math-tex"&gt; &lt;/span&gt;.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;dimension:&lt;/strong&gt;&nbsp;The dimension of the corresponding variety. Here always equal to 3.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;picard rank:&lt;/strong&gt;&nbsp;The rank of the Picard group. Here always equal to 1.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;acr gd matrix:&lt;/strong&gt;&nbsp;Grading matrix of the anticanonical ring w.r.t. to a minimal system of generators. The &lt;span class="math-tex"&gt; &lt;/span&gt;-th column is the degree of the&nbsp;&lt;span class="math-tex"&gt; &lt;/span&gt;-th generator.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;acr relations:&lt;/strong&gt;&nbsp;Minimal generating set of the ideal of relations of the anticanonical ring.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;hsNum, hsDenom:&lt;/strong&gt;&nbsp;Numerator and denominator of the Hilbert-Poincar&eacute; series, i.e.&nbsp;&lt;span class="math-tex"&gt; &lt;/span&gt;.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;hs0 - hs7:&lt;/strong&gt;&nbsp;The &lt;span class="math-tex"&gt; &lt;/span&gt;-th coefficient of the power series expansion of &lt;span class="math-tex"&gt; &lt;/span&gt;, i.e.&nbsp;&lt;span class="math-tex"&gt; &lt;/span&gt;&lt;/li&gt; &lt;li&gt;&lt;strong&gt;codim:&lt;/strong&gt;&nbsp;The codimension of the embedding into a weighted projective space provided by the choice of generators of the anticanonical ring.&lt;/li&gt; &lt;li&gt;&lt;strong&gt;genus:&lt;/strong&gt;&nbsp;This equals &lt;span class="math-tex"&gt; &lt;/span&gt;.&lt;/li&gt; &lt;/ul&gt;