Dataset Open Access

**The classification of toric canonical Fano 3-folds**

This dataset describes the classification of all toric canonical Fano 3-folds [1]. Equivalently, it describes the classification of all 3-dimensional convex lattice polytopes with exactly one interior lattice point.

A toric Fano 3-fold \(X\) (that is, a 3-dimensional toric variety with ample anticanonical divisor \(-K\)) with at worst canonical singularities corresponds to a 3-dimensional convex polytope \(P\) with vertices that are primitive integer vectors, and such that \(P\) contains exactly one lattice point, the origin, in its strict interior. The fan of \(X\) is given by the spanning fan of \(P\): that is, the fan whose cones are spanned by the faces of \(P\). In the language of toric geometry, \(P\) is in the lattice \(N \cong \mathbb{Z}^3\). Since two polytopes which are equal after a lattice change of basis give rise to isomorphic toric varieties, a polytope is regarded as being defined only up to the action of \(\mathrm{GL}(\mathbb{Z}^3)\). There are 674688 isomorphism classes.

For details, see the paper [1]. If you make use of this data, please consider citing [1] and the DOI for this data:

doi:10.5281/zenodo.5866330

**toricf3c.txt**

The file "toricf3c.txt" contains key:value records with keys and values as described below, where each record is separated by a blank line. Each key:value record determines a toric Fano 3-fold \(X\), or equivalently a lattice polytope \(P\), in the classification. There are 674688 records in the file.

**Example record**

id: 1

num_vertices: 16

num_faces: 10

num_points: 22

is_terminal: false

is_simplicial: false

is_regular: false

is_reflexive: false

vertex_list: [[2,1,1],[-1,0,-1],[-2,-1,-1],[0,1,1],[0,-1,-1],[0,-1,-2],[-2,-1,0],[0,1,0],[2,1,2],[-1,0,1],[1,1,2],[1,0,1],[1,1,0],[1,0,-1],[-1,-1,0],[-1,-1,-2]]

point_list: [[2,1,1],[-1,0,-1],[-2,-1,-1],[0,1,1],[0,-1,-1],[0,0,-1],[0,-1,-2],[-1,0,0],[-2,-1,0],[1,0,0],[0,1,0],[0,0,1],[-1,-1,-1],[2,1,2],[1,1,1],[-1,0,1],[1,1,2],[1,0,1],[1,1,0],[1,0,-1],[-1,-1,0],[-1,-1,-2]]

dual_list: [[0,-1,1],[0,-1,0],[1,-1,0],[-1,1,0],[0,1,0],[0,1,-1],[1/2,-1/2,-1/2],[-1/2,-1/2,1/2],[-1/2,3/2,-1/2],[1/2,-1/2,1/2]]

ehrhart_delta: [1,19,23,1]

hilbert_delta: [1,7,25,47,47,25,7,1]

normal_form: [[1,0,0],[0,1,0],[1,1,2],[0,-1,-2],[-1,0,-2],[1,-1,0],[-1,1,0],[1,0,2],[0,1,2],[-1,-1,-2],[0,-1,0],[-1,0,0],[-1,1,1],[1,1,1],[-1,-1,-3],[1,-1,1]]

volume: 44

degree: 10

gorenstein_index: 2

h1: 7

h2: 29

h3: 75

h4: 157

h5: 283

h6: 465

h7: 711

h8: 1033

h9: 1439

h10: 1941

e1: 23

e2: 109

e3: 303

e4: 649

e5: 1191

e6: 1973

e7: 3039

e8: 4433

e9: 6199

e10: 8381

picard_rank: 1

automorphism_order: 24

is_barycentre_zero: true

is_dual_barycentre_zero: true

We fix some notation. Let:

- \(N \cong \mathbb{Z}^3\)be a lattice of rank 3;
- \(P\) denote the lattice polytope in \(N_\mathbb{Q}=N\otimes_\mathbb{Z}\mathbb{Q}\) defined by the key:value record;
- \(F\) denote the spanning fan in N of P;
- \(X\) denote the toric Fano 3-fold corresponding to \(F\);
- \(M = \mathrm{Hom}(N,\mathbb{Z})\cong\mathbb{Z}^3\) denote the lattice dual to \(N\);
- \(P^*\) denote the (rational) polytope in \(M_\mathbb{Q}\) dual to \(P\) (also called the polar polytope);
- \(-K\) denote the anticanonical divisor of \(X\).

The keys and values are as follows.

id: A unique integer ID for this record, in the range 1 to 674688.

num_vertices: A positive integer. The number of vertices \(\#\mathrm{vert}(P)\) of \(P\). Equivalently, the number of rays of \(F\). By duality this is also equal to the number of 2-dimensional faces of \(P^*\).

num_faces: A positive integer. The number of 2-dimensional faces of \(P\). Equivalently, the number of top-dimensional cones of \(F\). By duality this is also equal to the number of vertices of \(P^*\).

num_points: A positive integer. The number of lattice points \(\#(P \cap N)\).

is_terminal: A boolean. True if and only if \(X\) has at worst terminal singularities. Equivalently, true if and only if the only lattice points on the boundary of \(P\) are the vertices of \(P\); that is, \(P \cap N = \mathrm{vert}(P) \cup \{0\}\).

is_simplicial: A boolean. True if and only if \(P\) is simplicial. Equivalently, true if and only if \(X\) is \(\mathbb{Q}\)-factorial. By duality, this is true if and only if \(P^*\) is simple.

is_regular: A boolean. True if and only if \(X\) is smooth. Equivalently, true if and only if every 2-dimensional face of \(P\) is a triangle whose vertices \(\mathbb{Z}\)-generate the lattice \(N\). If is_regular is true then both is_simplicial and is_terminal must be true.

is_reflexive: A boolean. True if and only if \(X\) is Gorenstein; that is, \(-K\) is Cartier. Equivalently, true if and only if \(P^*\) is a lattice polytope.

vertex_list: A sequence of lattice points in \(N\). The vertices \(\mathrm{vert}(P)\) of \(P\). Equivalently, the primitive lattice generators of the rays of \(F\). The number of points is given by num_vertices.

point_list: A sequence of lattice points in \(N\). The lattice points \(P \cap N\). The number of points is given by num_points.

dual_list: A sequence of rational points in \(M\). The vertices \(\mathrm{vert}(P^*)\) of \(P^*\). These will be lattice points if and only if is_reflexive is true. The number of points is given (via duality) by num_faces.

ehrhart_delta: A sequence \((1,a_1,a_2,1)\) of four integers, the first and last of which are always 1. This is the Ehrhart \(\delta\)-vector (or \(h^*\)-vector) of \(P\). The Ehrhart series \(\mathrm{Ehr}(P)\) of \(P\) is given by \(\mathrm{Ehr}(P) = (1 + a_1t + a_2t^2 + t^3) / (1 - t)^4\).

hilbert_delta: A sequence \((b_0,b_1,\ldots,b_N)\) of integers such that \(b_i = b_{N - i}\), and \(b_0 = b_N = 1\). This is called the Ehrhart \(\delta\)-vector (or \(h^*\)-vector) of \(P^*\). Write \(N = 4r - 1\). Then \(r\) is the quasiperiod of \(P^*\), and \(r\) divides gorenstein_index. The Ehrhart series of \(P^*\) is given by \(\mathrm{Ehr}(P^*) = (b_0 + b_1t + \ldots + b_Nt^N) / (1 - t^r)^4\). The Ehrhart series \(\mathrm{Ehr}(P^*)\) of \(P^*\) is equal to the Hilbert series \(\mathrm{Hilb}(X,-K)\).

normal_form: A sequence of lattice points in \(N\). The PALP normal form of the vertices of \(P\); see [2,3].

volume: A positive integer. The lattice-normalised volume \(\mathrm{Vol}(P)\) of \(P\). This is equal to the sum of the ehrhart_delta: \(\mathrm{Vol}(P) = 1 + a_1 + a_2 + 1\).

degree: A positive integer. The anticanonical degree \((-K)^3\) of \(X\). Equivalently, the lattice-normalised volume \(\mathrm{Vol}(P^*)\) of \(P^*\).

gorenstein_index: A positive integer. The Gorenstein index of \(X\); that is, the smallest multiple \(m>0\) such that \(-mK\) is Cartier. Equivalently, the smallest multiple \(m>0\) such that \(mP^*\) is a lattice polytope. This is 1 if and only if is_reflexive is true.

h1,...,h10: Positive integers. The value hi \(=h_i\) is equal to the number of lattice points in the \(i\)-th dilation of \(P^*\), that is, \(h_i = \#(iP^* \cap M)\). Equivalently, \(h_i = h^0(X,-iK)\). The values \(h_i\) can also be obtained from \(\mathrm{Ehr}(P^*)\), or equivalently from \(\mathrm{Hilb}(X,-K)\), via the power-series expansion \((b_0 + b_1t + \ldots + b_Nt^N) / (1 - t^r)^4 = 1 + h_1t + h_2t^2 + h_3t^3 + \ldots\), where \((b_0,b_1,\ldots,b_N)\) is given by hilbert_delta.

e1,...,e10: Positive integers. The value ei \(=e_i\) is equal to the number of lattice points in the \(i\)-th dilation of \(P\), that is, \(e_i = \#(iP \cap N)\). In particular, \(e_1\) is equal to num_points. The values \(e_i\) can also be obtained from \(\mathrm{Ehr}(P)\) via the power-series expansion \((1 + a_1t + a_2t^2 + t^3) / (1 - t)^4 = 1 + e_1t + e_2t^2 + e_3t^3 + \ldots\), where \((1,a_1,a_2,1)\) is given by ehrhart_delta.

picard_rank: Positive integer. The rank of the Picard group of \(X\). When is_simplicial is true, this is equal to \(\#\mathrm{vert}(P) - 3\).

automorphism_order: Positive integer. The order of the automorphism group \(\mathrm{Aut}(P) \leq \mathrm{GL}(\mathbb{Z}^3)\) of \(P\).

is_barycentre_zero: A boolean. True if and only if the barycentre of \(P\) is equal to the origin in \(N\).

is_dual_barycentre_zero: A boolean. True if and only if the barycentre of \(P^*\) is equal to the origin in \(M\).

**toricf3c.sql**

The file "toricf3c.sql" contain an sqlite-formatted version of the data described above, and can be imported into an sqlite database via, for example:

`$ cat toricf3c.sql | sqlite3 toricf3c.db`

This can then be easily queried. For example:

```
$ sqlite3 toricf3c.db
> SELECT COUNT(*) FROM toricf3c;
674688
> SELECT vertex_list FROM toricf3c WHERE degree = 72;
[[-1,-4,-6],[1,0,0],[0,1,0],[0,0,1]]
[[-1,-1,-3],[1,0,0],[0,1,0],[0,0,1]]
```

**References**

[1] Alexander M. Kasprzyk. *Canonical toric Fano threefolds*. Canadian Journal of Mathematics, 62(6):1293–1309, 2010.

[2] Maximilian Kreuzer, Harald Skarke. *PALP, a package for analyzing lattice polytopes with applications to toric geometry*. Computer Phys. Comm., 157:87-106, 2004.

[3] Roland Grinis, Alexander M. Kasprzyk. *Normal forms of convex lattice polytopes*. arXiv:1301.6641 [math.CO], 2013

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