. The rays are generated by the images of the standard basis for
.
The first examples illustrate the defining data for three different weighted projective spaces.
i1 : PP4 = weightedProjectiveSpace {1,1,1,1};
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i2 : rays PP4
o2 = {{-1, -1, -1}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
o2 : List
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i3 : max PP4
o3 = {{0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}}
o3 : List
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i4 : dim PP4
o4 = 3
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i5 : X = weightedProjectiveSpace {1,2,3};
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i6 : rays X
o6 = {{-2, -3}, {1, 0}, {0, 1}}
o6 : List
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i7 : max X
o7 = {{0, 1}, {0, 2}, {1, 2}}
o7 : List
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i8 : dim X
o8 = 2
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i9 : ring X
o9 = QQ[x , x , x ]
0 1 2
o9 : PolynomialRing
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i10 : Y = weightedProjectiveSpace({1,2,2,3,4}, CoefficientRing => ZZ/32003, Variable => y);
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i11 : rays Y
o11 = {{-2, -2, -3, -4}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0,
-----------------------------------------------------------------------
1}}
o11 : List
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i12 : max Y
o12 = {{0, 1, 2, 3}, {0, 1, 2, 4}, {0, 1, 3, 4}, {0, 2, 3, 4}, {1, 2, 3, 4}}
o12 : List
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i13 : dim Y
o13 = 4
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i14 : ring Y
ZZ
o14 = -----[y , y , y , y , y ]
32003 0 1 2 3 4
o14 : PolynomialRing
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The grading of the total coordinate ring for weighted projective space is determined by the weights. In particular, the class group is
.
i15 : cl PP4
1
o15 = ZZ
o15 : ZZ-module, free
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i16 : degrees ring PP4
o16 = {{1}, {1}, {1}, {1}}
o16 : List
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i17 : cl X
1
o17 = ZZ
o17 : ZZ-module, free
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i18 : degrees ring X
o18 = {{1}, {2}, {3}}
o18 : List
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i19 : cl Y
1
o19 = ZZ
o19 : ZZ-module, free
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i20 : degrees ring Y
o20 = {{1}, {2}, {2}, {3}, {4}}
o20 : List
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