The weighted projective space associated to a list
{q0,…, qd }, where no
d-element subset of
q0,…, qd has a nontrivial common factor, is a normal toric variety built from a fan in
N = ℤd+1/ℤ(q0,…,qd). The rays are generated by the images of the standard basis for
ℤd+1 and the maximal cones in the fan correspond to the
d-element subsets of
{0,...,d}.
The first examples illustrate the defining data for three different weighted projective spaces.
PP4 = weightedProjectiveSpace {1,1,1,1}; |
rays PP4 |
max PP4 |
dim PP4 |
X = weightedProjectiveSpace {1,2,3}; |
rays X |
max X |
dim X |
Y = weightedProjectiveSpace {1,2,2,3,4}; |
rays Y |
max Y |
dim Y |
The grading of the total coordinate ring for weighted projective space is determined by the weights. In particular, the class group is
ℤ.
cl PP4 |
degrees ring PP4 |
cl X |
degrees ring X |
cl Y |
degrees ring Y |
A weighted projective space is always simplicial but is typically not smooth
isSimplicial PP4 |
isSmooth PP4 |
isSimplicial X |
isSmooth X |
isSimplicial Y |
isSmooth Y |