weightedProjectiveSpace q
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.
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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 $\ZZ$.
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