next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
NormalToricVarieties :: weightedProjectiveSpace

weightedProjectiveSpace -- make a weighted projective space

Synopsis

Description

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 d+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 .
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

See also

Ways to use weightedProjectiveSpace :