Classification of toric Fano 5-folds / data supplement
[arXiv:math.AG/0702890]

data supplements
M. Kreuzer and B. Nill:
Classification of toric Fano 5-folds




Toric Fano polytopes: There are 866 Fano polytopes in 5 dimensions and their polar polytopes (facet equations), 124 Fano polytopes in 4 dimensions, 18 Fano polytopes in 3 dimensions, 5 Fano polytopes in 2 dimensions.
The polar polytopes (and other data) can be computed with the command poly.x -Dv of the software package PALP. (Vertices a of the polar polytope correspond to coefficient vectors for the bounding halfspace equations a.x+1≥0.)

NEF partitions and Complete Intersection Calabi-Yau 3-folds:

The 5d-Fano polytopes have between 3 and 159 nef partitions per polytope, which can be used to construct 36233 toric CICY 3-folds with 421 different pairs of Hodge numbers.


HOWTO compute these results with PALP: Making reflexive polytopes:
echo -e "3 1 1 1\n4 1 1 2\n6 1 2 3\n2 1 1 0 0 2 0 0 1 1" > 2d.cws class.x -po 2d.bin 2d.cws # 2d reflexive (binary) class.x -b -pi 2d.bin > 2d.ref # 2d reflexive (asci) echo "4 1 1 1 1 /Z2: 0 1 0 1" > 3d.cws ; cws.x -w3 >> 3d.cws ; cws.x -c3 >> 3d.cws class.x -po 3d.bin 3d.cws ; class.x -b -pi 3d.bin > 3d.ref # 3d reflexive (asci) http://quark.itp.tuwien.ac.at/~kreuzer/d4/ # 4d reflexive database (4.4 GB) The following is still under construction / verification: Making Fanos: This requires palp-1.1+fano.tar.gz (code will be contained in releases > palp-1.1) poly.x -U1e 2d.ref| poly.x -fN| poly.x -fv> ToricFano.2d # Result 2d: #Fano polytopes = 5 poly.x -U5 2d.ref 3d.fanocand # redundant list of lifts poly.x -DU1e 3d.fanocand | class.x -f -a -po 3d.fano.bin # Result 3d: #Fano polytopes = 18 class.x -b -pi 3d.fano.bin > ToricFano.3d # conversion to asci poly.x -U1v 3d.ref | grep -c P # test poly.x -U5 3d.ref 4d.fanocand # redundant list of lifts poly.x -DU1e 4d.fanocand | class.x -f -a -po 4d.fano.bin # Result 4d: #Fano polytopes = 124 class.x -b -pi 4d.fano.bin > ToricFano.4d # conversion to asci class.x -b -di d4/zzdb | poly.x -fU5 5d.fanocand # redundant list of lifts poly.x -DU1e 5d.fanocand | class.x -f -a -po 5d.fano.bin # Result 5d: #Fano polytopes = 866 class.x -b -pi 5d.fano.bin > ToricFano.5d # conversion to asci Computing NEF partitions nef.x -N -D -P ToricFanos.5d FanoCICY.cd2 # NEF partitions and CICYs