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

Usage:

ChowRing R
Inputs:
- R, a ring, the graded coordinate ring of the product of projective spaces \PP^{n_1}\times \cdots \times \PP^{n_m}
Outputs:
- a quotient ring, the Chow ring A=\ZZ[h_1,...,h_m]/(h_1^{n_1+1},...,h_m^{n_m+1}) of a product of projective spaces \PP^{n_1}\times \cdots \times\PP^{n_m}

Description

This method computes the Chow ring A=\ZZ[h_1,...,h_m]/(h_1^{n_1+1},...,h_m^{n_m+1}) of a product of projective spaces \PP^{n_1}\times \cdots \times\PP^{n_m}. 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 \PP^3x\PP^4.

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:

ChowRing(Ring)
ChowRing(Ring,Symbol)

For the programmer

The object ChowRing is a method function.

The source of this document is in CharacteristicClasses.m2:1855:0.