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NormalToricVarieties :: toricProjectiveSpace(ZZ)

toricProjectiveSpace(ZZ) -- 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, e2, …,ed of d together with vector -e1-e2-…-ed. The maximal cones in the fan correspond to the d-element subsets of {0,1, …,d}.

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

i1 : PP1 = toricProjectiveSpace 1;
i2 : rays PP1

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

o2 : List
i3 : max PP1

o3 = {{0}, {1}}

o3 : List
i4 : dim PP1

o4 = 1
i5 : ring PP1

o5 = QQ[x , x ]
         0   1

o5 : PolynomialRing
i6 : ideal PP1

o6 = ideal (x , x )
             1   0

o6 : Ideal of QQ[x , x ]
                  0   1
i7 : assert (isSmooth PP1 and isComplete PP1)
i8 : PP3 = toricProjectiveSpace (3, CoefficientRing => ZZ/32003, Variable => y);
i9 : rays PP3

o9 = {{-1, -1, -1}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}

o9 : List
i10 : max PP3

o10 = {{0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2, 3}}

o10 : List
i11 : dim PP3

o11 = 3
i12 : ring PP3

        ZZ
o12 = -----[y , y , y , y ]
      32003  0   1   2   3

o12 : PolynomialRing
i13 : ideal PP3

o13 = ideal (y , y , y , y )
              3   2   1   0

                 ZZ
o13 : Ideal of -----[y , y , y , y ]
               32003  0   1   2   3
i14 : assert (isSmooth PP3 and isComplete PP3)

See also