A normal toric variety is simplical if every cone in its fan is simplicial and a cone is simplicial if its minimal generators are linearly independent over
. In fact, the following conditions on a normal toric variety
Given a normal toric variety, this method makes a simplicial toric variety with the same rays by triangulating the non-simplicial maximal cones. For the
strategy, the triangulation is constructed by repeated regular subdivisions using random integral weight vectors. For the
strategy, the triangulation is constructed by repeated pushing subdivisions (i.e. blowups at a given ray).
i1 : X = normalToricVariety(id_(ZZ^3) | - id_(ZZ^3));
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i2 : isSimplicial X
o2 = false
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i3 : Y1 = makeSimplicial X;
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i4 : isSimplicial Y1
o4 = true
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i5 : rays Y1 === rays X
o5 = true
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i6 : max Y1
o6 = {{0, 1, 2}, {0, 1, 5}, {0, 2, 6}, {0, 4, 5}, {0, 4, 6}, {1, 2, 3}, {1,
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3, 7}, {1, 5, 7}, {2, 3, 7}, {2, 6, 7}, {4, 5, 7}, {4, 6, 7}}
o6 : List
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i7 : max X
o7 = {{0, 1, 2, 3}, {0, 1, 4, 5}, {0, 2, 4, 6}, {1, 3, 5, 7}, {2, 3, 6, 7},
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{4, 5, 6, 7}}
o7 : List
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i8 : Y2 = makeSimplicial(X, Strategy => 1);
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i9 : isSimplicial Y2
o9 = true
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i10 : rays Y2 === rays X
o10 = true
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i11 : max Y2
o11 = {{0, 1, 3}, {0, 1, 5}, {0, 2, 3}, {0, 2, 6}, {0, 4, 5}, {0, 4, 6}, {1,
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3, 7}, {1, 5, 7}, {2, 3, 7}, {2, 6, 7}, {4, 5, 7}, {4, 6, 7}}
o11 : List
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If the initial toric variety is simplicial, then this method simply returns it.