## Reflexive Gorenstein Cones

• In arXiv:1204.1181 a classification scheme for reflexive Gorenstein cones was presented. An essential role for that scheme is played by basic weight systems, i.e. weight systems containing no weights of 1/2 or 1 and no two weights whose sum is 1.
We use the following notation:
• d ... the dimension of the cone,
• r ... the index of the cone,
• dCY = d - 2r ... the "Calabi-Yau-dimension" of the cone,
• qi = ni/k ... the weights.

• The files presented below contain lines of the form
```k n1 ... nd M: # # N: # # H: ...
```
if the cone determined by the weight system is reflexive, and of the form
```k n1 ... nd M: # # F: #
```
otherwise, where
• k n1 ... nd encodes the weights as qi = ni/k,
• M: # # gives the numbers of lattice points and vertices of the support of the cone,
• N: # # gives the numbers of lattice points and vertices of the support of the dual cone,
• H: ... gives the "stringy Hodge numbers" in the format
h11 ... h1,dCY-1 [Euler number] (non-standard h0i),
where "non-standard h0i" indicates any h0i's that do not follow the standard pattern
h00 = hdCYdCY = 1, h0i = 0 for 0 < i < dCY,
• F: # gives the number of facets of the support of the cone.
Weight systems for (d,r) with half-integer r are listed as the weight systems of type (d+1,r+1/2) that are obtained by adding one weight of 1/2; e.g. (1/6, 1/6, 1/6) of type (3, 1/2) becomes (1/6, 1/6, 1/6, 1/2) of type (4, 1) etc.

• The basic weight systems for dCY = 2 are listed in the following files: The first 95 = 48 + 47 correspond to K3 hypersurfaces.
Each of the weight systems gives rise to a reflexive Gorenstein cone.

• The basic weight systems for dCY = 3 are listed in the following files: The first 184,026 = 97,036 + 86,990 correspond to Calabi-Yau hypersurfaces, and also to reflexive Gorenstein cones of index 1.
The cases for d > 5 do not necessarily give rise to reflexive Gorenstein cones. The sublists of reflexive cones are