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CharacteristicClasses :: ChowRing

ChowRing -- Computes the Chow ring of a product of projective spaces m projective spaces given the coordinate ring

Synopsis

Description

This method computes the Chow ring A=ℤ[h1,...,hm]/(h1n1+1,...,hmnm+1) of a product of projective spaces ℙn1×…×ℙnm. It is needed for input into the methods Segre, Chern and CSM to ensure that these methods return results in the same ring. We give an example of the use of this method to work with elements of the Chow ring of ℙ3xℙ4.

i1 : R=MultiProjCoordRing({3,4})

o1 = R

o1 : PolynomialRing
i2 : A=ChowRing(R)

o2 = A

o2 : QuotientRing
i3 : I=ideal(random({1,0},R));

o3 : Ideal of R
i4 : K=ideal(random({1,1},R));

o4 : Ideal of R
i5 : c=Chern(A,I)

        3 4      3 3      2 4      3 2      2 3       4      3        2 2  
o5 = 15h h  + 30h h  + 15h h  + 30h h  + 30h h  + 5h h  + 15h h  + 30h h  +
        1 2      1 2      1 2      1 2      1 2     1 2      1 2      1 2  
     ------------------------------------------------------------------------
          3     3      2          2     2
     10h h  + 3h  + 15h h  + 10h h  + 3h  + 5h h  + h
        1 2     1      1 2      1 2     1     1 2    1

o5 : A
i6 : s=Segre(A,K)

        3 4      3 3      2 4      3 2      2 3       4     3       2 2  
o6 = 35h h  - 20h h  - 15h h  + 10h h  + 10h h  + 5h h  - 4h h  - 6h h  -
        1 2      1 2      1 2      1 2      1 2     1 2     1 2     1 2  
     ------------------------------------------------------------------------
         3    4    3     2         2    3    2            2
     4h h  - h  + h  + 3h h  + 3h h  + h  - h  - 2h h  - h  + h  + h
       1 2    2    1     1 2     1 2    2    1     1 2    2    1    2

o6 : A
i7 : s-c

        3 4      3 3      2 4      3 2      2 3      3        2 2        3  
o7 = 20h h  - 50h h  - 30h h  - 20h h  - 20h h  - 19h h  - 36h h  - 14h h  -
        1 2      1 2      1 2      1 2      1 2      1 2      1 2      1 2  
     ------------------------------------------------------------------------
      4     3      2         2    3     2            2
     h  - 2h  - 12h h  - 7h h  + h  - 4h  - 7h h  - h  + h
      2     1      1 2     1 2    2     1     1 2    2    2

o7 : A
i8 : s*c

        3 4      3 3      2 4      3 2      2 3       4      3        2 2  
o8 = 12h h  + 21h h  + 12h h  + 20h h  + 19h h  + 4h h  + 10h h  + 15h h  +
        1 2      1 2      1 2      1 2      1 2     1 2      1 2      1 2  
     ------------------------------------------------------------------------
         3     3     2         2    2
     6h h  + 2h  + 6h h  + 4h h  + h  + h h
       1 2     1     1 2     1 2    1    1 2

o8 : A

We may also specify the variable to be used for the Chow ring.

i9 : A2=ChowRing(R,symbol v)

o9 = A2

o9 : QuotientRing
i10 : describe A2

      ZZ[v , v ]
          1   2
o10 = ----------
         4   5
       (v , v )
         1   2

Ways to use ChowRing :