A normal toric variety is simplicial if every cone in its fan is simplicial and a cone is simplicial if its minimal generators are linearly independent over $\QQ$. In fact, the following conditions on a normal toric variety $X$ are equivalent:
For more information, see Proposition 4.2.7 in Cox-Little-Schenck's Toric Varieties.
Projective spaces, weighted projective spaces, and Hirzebruch surfaces are simplicial.
i1 : PP1 = toricProjectiveSpace 1; |
i2 : assert (isSimplicial PP1 and isProjective PP1) |
i3 : FF7 = hirzebruchSurface 7; |
i4 : assert (isSimplicial FF7 and isProjective FF7) |
i5 : AA3 = affineSpace 3; |
i6 : assert (isSimplicial AA3 and not isComplete AA3 and # max AA3 === 1) |
i7 : P12234 = weightedProjectiveSpace {1,2,2,3,4}; |
i8 : assert (isSimplicial P12234 and isProjective P12234) |
i9 : U = normalToricVariety ({{4,-1},{0,1}},{{0,1}}); |
i10 : assert (isSimplicial U and not isSmooth U) |
However, not all normal toric varieties are simplicial.
i11 : Q = normalToricVariety ({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}}); |
i12 : assert (not isSmooth Q and not isSimplicial Q and not isComplete Q) |
i13 : Y = normalToricVariety ( id_(ZZ^3) | - id_(ZZ^3)); |
i14 : assert (not isSimplicial Y and isProjective Y) |
To avoid repeating a computation, the package caches the result in the CacheTable of the normal toric variety.