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

## Synopsis

• Function: isSimplicial
• Usage:
isSimplicial X
• Inputs:
• X, ,
• Outputs:
• , that is true if the minimal nonzero lattice points on the rays in each maximal cone in the associated fan of form part of a $\QQ$-basis

## Description

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.

 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)
 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)