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,
  • d_CY = d - 2r ... the "Calabi-Yau-dimension" of the cone,
  • q_i = n_i/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 ... n_d encodes the weights as q_i = n_i/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 ... h_1,dCY-1 [Euler number] (non-standard h_0i),
    where "non-standard h_0i" indicates any h_0i's that do not follow the standard pattern
    h₀₀ = h_dCYdCY = 1, h_0i = 0 for 0 < i < d_CY,
  • 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 d_CY = 2 are listed in the following files:
  • d=3, r=1/2 (48)
  • d=4, r=1 (47)
  • d=5, r=3/2 (28)
  • d=6, r=2 (1)
  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 d_CY = 3 are listed in the following files:
  • d=4, r=1/2 (97,036)
  • d=5, r=1 (86,990)
  • d=6, r=3/2 (168,107)
  • d=7, r=2 (34,256)
  • d=8, r=5/2 (6,066)
  • d=9, r=3 (1)
  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
  • d=6, r=3/2 (112,817)
  • d=7, r=2 (18,962)
  • d=8, r=5/2 (1,321)
  • d=9, r=3 (1)