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

projectiveSpace -- Makes an AbstractVariety representing projective space

Synopsis

Description

Constructs the projective space P of 1-quotients of the trivial bundle on the base variety baseVariety. The Chow ring is set to be the polynomial ring over the Chow ring of baseVariety, with variable h. The tangent bundle of X is available as an AbstractSheaf, accessed by X.TangentBundle. Here baseVariety and VariableName are optional.
i1 : P=projectiveSpace(3)

o1 = P

o1 : a flag bundle with ranks {3, 1}
i2 : todd P

              11 2    3
o2 = 1 + 2h + --h  + h
               6

               QQ[][H   , H   , H   , h]
                     1,1   1,2   1,3
o2 : ---------------------------------------------
     (H    + h, H    + H   h, H    + H   h, H   h)
       1,1       1,2    1,1    1,3    1,2    1,3
i3 : chi(OO_P(3))

o3 = 20

o3 : QQ[]
If we want a projective space where we can compute the Hilbert Polynomial of a sheaf, we need a variable to represent an integer. We define a base variety that is a point pt containing this variable.
i4 : pt = base(n)

o4 = pt

o4 : an abstract variety of dimension 0
i5 : Q=projectiveSpace(4,pt, VariableName => h)

o5 = Q

o5 : a flag bundle with ranks {4, 1}
i6 : chi(OO_Q(n))

      1 4    5 3   35 2   25
o6 = --n  + --n  + --n  + --n + 1
     24     12     24     12

o6 : QQ[n]
If be build a projective space over another variety, the dimensions add:
i7 : baseVariety = projectiveSpace(4, VariableName => h)

o7 = baseVariety

o7 : a flag bundle with ranks {4, 1}
i8 : P = projectiveSpace (3,baseVariety, VariableName => H)

o8 = P

o8 : a flag bundle with ranks {3, 1}
i9 : dim P

o9 = 7
i10 : todd P

                5      11 2          35 2      3   55   2   35 2    25 3   
o10 = 1 + (2H + -h) + (--H  + 5h*H + --h ) + (H  + --h*H  + --h H + --h ) +
                2       6            12            12        6      12     
      -----------------------------------------------------------------------
       5   3   385 2 2   25 3     4     35 2 3   275 3 2     4      25 3 3  
      (-h*H  + ---h H  + --h H + h ) + (--h H  + ---h H  + 2h H) + (--h H  +
       2        72        6             12        72                12      
      -----------------------------------------------------------------------
      11 4 2     4 3
      --h H ) + h H
       6

                    QQ[][H   , H   , H   , H   , h]
                          1,1   1,2   1,3   1,4
      -----------------------------------------------------------[H   , H   , H   , H]
      (H    + h, H    + H   h, H    + H   h, H    + H   h, H   h)  1,1   1,2   1,3
        1,1       1,2    1,1    1,3    1,2    1,4    1,3    1,4
o10 : --------------------------------------------------------------------------------
                        (H    + H, H    + H   H, H    + H   H, H   H)
                          1,1       1,2    1,1    1,3    1,2    1,3

Ways to use projectiveSpace :