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NormalToricVarieties :: Chow ring

Chow ring -- make the rational Chow ring

Synopsis

The rational Chow ring of a variety $X$ is an associative commutative ring graded by codimension. The $k$-th graded component of this ring is the rational vector space spanned by the rational equivalence classes of subvarieties in $X$ having codimension $k$. For generically transverse subvarieties $Y$ and $Z$ in $X$, the product satisfies $[Y][Z] = [Y \cap Z]$.

For a complete simplicial normal toric variety, the rational Chow ring has an explicit presentation. Specifically, it is the quotient of the polynomial ring with variables indexed by the set of rays in the underlying fan by the sum of two ideals. The first ideal is the Stanley-Reisner ideal of the fan (equivalently the Alexander dual of the irrelevant ideal) and the second ideal is generated by linear forms that encode the rays(NormalToricVariety) in the fan. In this context, the rational Chow ring is isomorphic to the rational cohomology of $X$.

The rational Chow ring of projective space is generated by the rational equivalence class of a hyperplane.

i1 : PP3 = toricProjectiveSpace 3;
i2 : A0 = intersectionRing PP3

o2 = A0

o2 : QuotientRing
i3 : assert (# rays PP3 === numgens A0)
i4 : ideal A0

o4 = ideal (t t t t , - t  + t , - t  + t , - t  + t )
             0 1 2 3     0    1     0    2     0    3

o4 : Ideal of QQ[][t ..t ]
                    0   3
i5 : dual monomialIdeal PP3 + ideal ((vars ring PP3) * matrix rays PP3)

o5 = ideal (x x x x , - x  + x , - x  + x , - x  + x )
             0 1 2 3     0    1     0    2     0    3

o5 : Ideal of QQ[x ..x ]
                  0   3
i6 : minimalPresentation A0

     QQ[t ]
         3
o6 = ------
        4
       t
        3

o6 : QuotientRing
i7 : for i to dim PP3 list hilbertFunction (i, A0)

o7 = {1, 1, 1, 1}

o7 : List

The rational Chow ring for the product of two projective spaces is the tensor product of the rational Chow rings of the factors.

i8 : X = toricProjectiveSpace (2) ** toricProjectiveSpace (3);
i9 : A1 = intersectionRing X

o9 = A1

o9 : QuotientRing
i10 : assert (# rays X === numgens A1)
i11 : ideal A1

o11 = ideal (t t t , t t t t , - t  + t , - t  + t , - t  + t , - t  + t , -
              0 1 2   3 4 5 6     0    1     0    2     3    4     3    5   
      -----------------------------------------------------------------------
      t  + t )
       3    6

o11 : Ideal of QQ[][t ..t ]
                     0   6
i12 : minimalPresentation A1

      QQ[t , t ]
          2   6
o12 = ----------
         3   4
       (t , t )
         2   6

o12 : QuotientRing
i13 : for i to dim X list hilbertFunction (i, A1)

o13 = {1, 2, 3, 3, 2, 1}

o13 : List

We end with a slightly larger example.

i14 : Y = time smoothFanoToricVariety(5,100);
     -- used 0.371885 seconds
i15 : A2 = intersectionRing Y;
i16 : assert (# rays Y === numgens A2)
i17 : ideal A2

o17 = ideal (t t , t t , t t , t t , t t , t t , t t , t t , t t t  ,
              2 3   2 5   4 5   3 6   4 6   1 7   7 9   8 9   0 1 10 
      -----------------------------------------------------------------------
      t t t  , - t  + t  , - t  - t  + t  , t  - t  - t  + t , t  - t  - t ,
       0 8 10     0    10     1    8    10   2    3    4    6   4    5    6 
      -----------------------------------------------------------------------
      t  + t  - t  - 2t  )
       7    8    9     10

o17 : Ideal of QQ[][t ..t  ]
                     0   10
i18 : minimalPresentation A2

                                                              QQ[t , t ..t , t ..t  ]
                                                                  3   5   6   8   10
o18 = ---------------------------------------------------------------------------------------------------------------------------------------
        2                 2   2                       2   2                             2             2                      2     3      2
      (t  + t t , t t  + t , t  + t t , t t , t t  + t , t  - t t  - 3t t   + t t   + 2t  , - t t  + t  + 2t t  , t t , - t t   + t  , t t  )
        3    3 5   3 5    5   5    5 6   3 6   5 6    6   8    8 9     8 10    9 10     10     8 9    9     9 10   8 9     8 10    10   8 10

o18 : QuotientRing
i19 : for i to dim Y list time hilbertFunction (i, A2)
     -- used 0.00134633 seconds
     -- used 0.00107529 seconds
     -- used 0.000935664 seconds
     -- used 0.000911619 seconds
     -- used 0.000894752 seconds
     -- used 0.000949369 seconds

o19 = {1, 6, 13, 13, 6, 1}

o19 : List

See also