As a an example, let’s take the 101th example on this list.
i1 : topes = kreuzerSkarkeDim3(); |
i2 : #topes o2 = 4319 |
i3 : tope = topes_100 o3 = 3 5 M:12 5 N:15 5 Pic:13 Cor:4 id:100 1 0 0 -3 1 0 1 0 0 -2 0 0 1 -2 2 o3 : KSEntry |
i4 : header = description tope o4 = 3 5 M:12 5 N:15 5 Pic:13 Cor:4 id:100 |
i5 : A = matrix tope o5 = | 1 0 0 -3 1 | | 0 1 0 0 -2 | | 0 0 1 -2 2 | 3 5 o5 : Matrix ZZ <--- ZZ |
The first line gives some information about the example, see Kreuzer-Skarke description headers for more details. The polytope is the convex hull of the columns of the matrix A.
One can use the packages Polyhedra and NormalToricVarieties to investigate these polyhedra, and the associated toric varieties.
i6 : needsPackage "Polyhedra" o6 = Polyhedra o6 : Package |
i7 : P = convexHull A o7 = P o7 : Polyhedron |
i8 : P2 = polar P o8 = P2 o8 : Polyhedron |
i9 : # latticePoints P o9 = 12 |
i10 : # latticePoints P2 o10 = 15 |
i11 : # vertices P o11 = 5 |
i12 : # vertices P2 o12 = 5 |
i13 : isReflexive P o13 = true |
i14 : needsPackage "NormalToricVarieties" o14 = NormalToricVarieties o14 : Package |
i15 : V0 = normalToricVariety normalFan P o15 = V0 o15 : NormalToricVariety |
i16 : dim V0 o16 = 3 |
i17 : max V0 o17 = {{0, 1, 2}, {0, 1, 4}, {0, 2, 3, 4}, {1, 2, 3}, {1, 3, 4}} o17 : List |
i18 : rays V0 o18 = {{1, 0, -1}, {-1, -1, -1}, {1, -1, -1}, {-1, -1, 2}, {-1, 2, 2}} o18 : List |
i19 : V = makeSimplicial V0 o19 = V o19 : NormalToricVariety |
i20 : isSimplicial V o20 = true |
i21 : isProjective V o21 = true |
i22 : isSmooth V o22 = false |
i23 : dim V o23 = 3 |