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

projectiveSpace -- make a projective space

Synopsis

Description

Projective d-space is a smooth complete normal toric variety. The rays are generated by the standard basis e1,…,ed of d together with -e1-…-ed. The maximal cones in the fan correspond to the d-element subsets of {0,...,d}.

The examples illustrate the projective line and projective 3-space.

i1 : PP1 = projectiveSpace 1;
i2 : rays PP1

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

o2 : List
i3 : max PP1

o3 = {{0}, {1}}

o3 : List
i4 : dim PP1

o4 = 1
i5 : isComplete PP1

o5 = true
i6 : isSmooth PP1

o6 = true
i7 : ring PP1

o7 = QQ[x , x ]
         0   1

o7 : PolynomialRing
i8 : ideal PP1

o8 = ideal (x , x )
             1   0

o8 : Ideal of QQ[x , x ]
                  0   1
i9 : PP3 = projectiveSpace 3;
i10 : rays PP3

o10 = {{-1, -1, -1}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}

o10 : List
i11 : max PP3

o11 = {{0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}}

o11 : List
i12 : dim PP3

o12 = 3
i13 : isComplete PP3

o13 = true
i14 : isSmooth PP3

o14 = true
i15 : ring PP3

o15 = QQ[x , x , x , x ]
          0   1   2   3

o15 : PolynomialRing
i16 : ideal PP3

o16 = ideal (x , x , x , x )
              3   2   1   0

o16 : Ideal of QQ[x , x , x , x ]
                   0   1   2   3

See also

Ways to use projectiveSpace :