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.

Description:

The columns of the csv-file are labelled as follows:

ID: The ID of the family within the database, numbered 1 through 538.

case: Refers to the corresponding case in Proposition 2.24. Numbered 1 through 8.

format: The format of the family as defined on page 13.

generator matrix: The defining matrix \(P\) as defined in Construction 2.2.

class group: The isomorphy type of the class group. Presented as \([0, t_1, \dots, t_n]\) meaning \(\mathbb{Z} \oplus ( \mathbb{Z}/t_1 \mathbb{Z} \oplus \dots \oplus \mathbb{Z}/t_n \mathbb{Z} )\).

gd matrix: Grading matrix of the Cox ring. The \(i\)-th column is the degree of the \(i\)-th Cox ring generator.

relations: Minimal generating set of the ideal of relations of the Cox ring.

relation degree: Degree of the minimal generators of the ideal of relations.

anticanonical class: The anticanonical class.

degree: Anticanonical self-intersection number \(-\mathcal{K}_X^3\).

gorenstein index: The smallest positive \(\iota \in \mathbb{Z}\) such that \(\iota \mathcal{K}_X\) is Cartier. Here always equal to 1.

fano index: The largest \(k \in \mathbb{Z}\) such that \(k D = \mathcal{K}_X\) for some Cartier divisor \(D\).

dimension: The dimension of the corresponding variety. Here always equal to 3.

picard rank: The rank of the Picard group. Here always equal to 1.

acr gd matrix: 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.

acr relations: Minimal generating set of the ideal of relations of the anticanonical ring.

hsNum, hsDenom: Numerator and denominator of the Hilbert-Poincaré series, i.e. \({\rm HP}_X(t) = {\rm hsNum}(t)/ {\rm hsDenom}(t)\).

hs0 - hs7: The \(i\)-th coefficient of the power series expansion of \({\rm HP}_X(t)\), i.e. \({\rm HP}_X(t) = {\rm hs0} + {\rm hs1}\, t + {\rm hs2}\, t^2 + \dots + {\rm hs7}\, t^7 + \dots\)

codim: The codimension of the embedding into a weighted projective space provided by the choice of generators of the anticanonical ring.

genus: This equals \(h^0 (X, -\mathcal{K}_X)-2\).