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NormalToricVarieties :: isSimplicial(NormalToricVariety)

isSimplicial(NormalToricVariety) -- whether a normal toric variety is simplicial



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:

  • X is simplicial,
  • every torus-invariant Weil divisor on X has a positive integer multiple that is Cartier,
  • the Picard group of X has finite index in the class group of X,
  • X has only finite quotient singularities.

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.

See also