The weighted projective space associated to a list $\{ q_0, q_1, \dots, q_d \}$, where no $d$-element subset of $q_0, q_1, \dots, q_d$ has a nontrivial common factor, is a projective simplicial normal toric variety built from a fan in $N = \ZZ^{d+1}/\ZZ(q_0, q_1, \dots,q_d)$. The rays are generated by the images of the standard basis for $\ZZ^{d+1}$, and the maximal cones in the fan correspond to the $d$-element subsets of $\{ 0, 1, ..., d \}$. A weighted projective space is typically not smooth.
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 |
i5 : assert (isWellDefined PP4 and isProjective PP4 and isSmooth PP4) |
i6 : X = weightedProjectiveSpace {1,2,3};
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i7 : rays X
o7 = {{-2, -3}, {1, 0}, {0, 1}}
o7 : List
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i8 : max X
o8 = {{0, 1}, {0, 2}, {1, 2}}
o8 : List
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i9 : dim X o9 = 2 |
i10 : ring X
o10 = QQ[x ..x ]
0 2
o10 : PolynomialRing
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i11 : assert (isWellDefined X and isProjective X and isSimplicial X and not isSmooth X) |
i12 : Y = weightedProjectiveSpace ({1,2,2,3,4}, CoefficientRing => ZZ/32003, Variable => y);
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i13 : rays Y
o13 = {{-2, -2, -3, -4}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0,
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1}}
o13 : List
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i14 : max Y
o14 = {{0, 1, 2, 3}, {0, 1, 2, 4}, {0, 1, 3, 4}, {0, 2, 3, 4}, {1, 2, 3, 4}}
o14 : List
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i15 : dim Y o15 = 4 |
i16 : ring Y
ZZ
o16 = -----[y ..y ]
32003 0 4
o16 : PolynomialRing
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i17 : assert (isWellDefined Y and isProjective Y and isSimplicial Y and not isSmooth Y) |
The grading of the total coordinate ring for weighted projective space is determined by the weights. In particular, the class group is $\ZZ$.
i18 : classGroup PP4
1
o18 = ZZ
o18 : ZZ-module, free
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i19 : degrees ring PP4
o19 = {{1}, {1}, {1}, {1}}
o19 : List
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i20 : classGroup X
1
o20 = ZZ
o20 : ZZ-module, free
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i21 : degrees ring X
o21 = {{1}, {2}, {3}}
o21 : List
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i22 : classGroup Y
1
o22 = ZZ
o22 : ZZ-module, free
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i23 : degrees ring Y
o23 = {{1}, {2}, {2}, {3}, {4}}
o23 : List
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