Dataset Open Access

# The classification of toric canonical Fano 3-folds

Kasprzyk, Alexander

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

Files (1.4 GB)
Name Size
md5:65d3616852dbf7b1a6d4b53b00626032
7.0 kB
md5:3366cba82226376d441b8bafdbf61e74
8.4 kB
toricf3c.sql
md5:d38e5d87917a3444a8b5a877e1868f00
563.7 MB
toricf3c.txt
md5:fd9a5782c80f799892d8cd914eb9d43c
817.1 MB
• Kasprzyk, A. (2010). Canonical Toric Fano Threefolds. Canadian Journal of Mathematics, 62(6), 1293-1309. doi:10.4153/CJM-2010-070-3

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