# weightedProjectiveSpace(List) -- make a weighted projective space

## Synopsis

• Function: weightedProjectiveSpace
• Usage:
weightedProjectiveSpace q
• Inputs:
• q, a list, of relatively prime positive integers
• Optional inputs:
• CoefficientRing => a ring, default value QQ, that specifies the coefficient ring of the total coordinate ring
• Variable => , default value x, that specifies the base name for the indexed variables in the total coordinate ring
• Outputs:
• , that is a weighted projective space

## 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