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NormalToricVarieties :: weightedProjectiveSpace(List)

weightedProjectiveSpace(List) -- make a weighted projective space

Synopsis

Description

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};
i2 : rays PP4

o2 = {{-1, -1, -1}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}

o2 : List
i3 : max PP4

o3 = {{0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}}

o3 : List
i4 : dim PP4

o4 = 3
i5 : assert (isWellDefined PP4 and isProjective PP4 and isSmooth PP4)
i6 : X = weightedProjectiveSpace {1,2,3};
i7 : rays X

o7 = {{-2, -3}, {1, 0}, {0, 1}}

o7 : List
i8 : max X

o8 = {{0, 1}, {0, 2}, {1, 2}}

o8 : List
i9 : dim X

o9 = 2
i10 : ring X

o10 = QQ[x ..x ]
          0   2

o10 : PolynomialRing
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);
i13 : rays Y

o13 = {{-2, -2, -3, -4}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0,
      -----------------------------------------------------------------------
      1}}

o13 : List
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
i15 : dim Y

o15 = 4
i16 : ring Y

        ZZ
o16 = -----[y ..y ]
      32003  0   4

o16 : PolynomialRing
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
i19 : degrees ring PP4

o19 = {{1}, {1}, {1}, {1}}

o19 : List
i20 : classGroup X

        1
o20 = ZZ

o20 : ZZ-module, free
i21 : degrees ring X

o21 = {{1}, {2}, {3}}

o21 : List
i22 : classGroup Y

        1
o22 = ZZ

o22 : ZZ-module, free
i23 : degrees ring Y

o23 = {{1}, {2}, {2}, {3}, {4}}

o23 : List

See also