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CharacteristicClasses :: CheckToricVarietyValid

CheckToricVarietyValid -- Checks if the input normal toric variety X is a valid choice for an ambient space when computing characteristic classes of subschemes V of X

Synopsis

Description

Note that if you are working with subvarieites of some product of projective spaces ℙn1×…×ℙnm then the ambient space is a valid choice for use with the ChacteristicsClasses package and there is no need to load the NormalToricVarieties Package or to check validity. For other cases the CheckToricVarietyValid method returns true if the input toric variety X may be used as an ambient space for other characteristic class computations, i.e. if this method returns true we may use methods such as CSM(X,I), Chern(X,I) and Segre(X,I) for I an ideal in the coordinate ring of X. We will see an example of a valid toric variety which is not a product of projective spaces and a smooth toric variety which is not valid.

i1 : needsPackage "NormalToricVarieties"

o1 = NormalToricVarieties

o1 : Package
i2 : Rho = {{1,0,0},{0,1,0},{0,0,1},{-1,-1,0},{0,0,-1}}

o2 = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {-1, -1, 0}, {0, 0, -1}}

o2 : List
i3 : Sigma = {{0,1,2},{1,2,3},{0,2,3},{0,1,4},{1,3,4},{0,3,4}}

o3 = {{0, 1, 2}, {1, 2, 3}, {0, 2, 3}, {0, 1, 4}, {1, 3, 4}, {0, 3, 4}}

o3 : List
i4 : X = normalToricVariety(Rho,Sigma,CoefficientRing =>ZZ/32749)

o4 = X

o4 : NormalToricVariety
i5 : CheckToricVarietyValid(X)

o5 = true
i6 : R=ring(X)

o6 = R

o6 : PolynomialRing
i7 : I=ideal(R_0^4*R_1,R_0*R_3*R_4*R_2-R_2^2*R_0^2)

             4       2 2
o7 = ideal (x x , - x x  + x x x x )
             0 1     0 2    0 2 3 4

o7 : Ideal of R
i8 : Segre(X,I)

          2       2
o8 = - 72x x  + 3x  + 8x x  + x
          3 4     3     3 4    3

               ZZ[x , x , x , x , x ]
                   0   1   2   3   4
o8 : -----------------------------------------
     (x x , x x x , x  - x , x  - x , x  - x )
       2 4   0 1 3   0    3   1    3   2    4
i9 : W = smoothFanoToricVariety(4,123)

o9 = W

o9 : NormalToricVariety
i10 : CheckToricVarietyValid(W)

o10 = false

Even if we can not preform computations on subschemes we may still compute the CSM class of the toric variety itself using the CSM command.

i11 : Ch=ToricChowRing W

o11 = Ch

o11 : QuotientRing
i12 : CSM W

           4       2       2        2        2      3     2     2     2  
o12 = - 24x  + 5x x  + 5x x  + 10x x  + 10x x  - 10x  + 3x  + 3x  - 3x  +
           8     2 8     5 8      6 8      7 8      8     2     5     6  
      -----------------------------------------------------------------------
               2                                   2
      x x  - 3x  + 9x x  + 9x x  + x x  + x x  + 6x  + 3x  + 3x  - x  - x  +
       6 7     7     2 8     5 8    6 8    7 8     8     2     5    6    7  
      -----------------------------------------------------------------------
      5x  + 1
        8

                                                                                              ZZ[x , x , x , x , x , x , x , x , x ]
                                                                                                  0   1   2   3   4   5   6   7   8
o12 : ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
      (x x , x x , x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x  - x  + x  - x , x  - x  + x  - x , x  - x  + x  - x , x  - x  + x  - x )
        0 3   1 4   2 5   0 1 2   0 1 8   0 2 7   0 7 8   1 2 6   1 6 8   2 6 7   3 4 5   3 4 8   3 5 7   3 7 8   4 5 6   4 6 8   5 6 7   6 7 8   0    2    6    8   1    2    7    8   3    5    6    8   4    5    7    8
i13 : CSM(Ch,W)

           4       2       2        2        2      3     2     2     2  
o13 = - 24x  + 5x x  + 5x x  + 10x x  + 10x x  - 10x  + 3x  + 3x  - 3x  +
           8     2 8     5 8      6 8      7 8      8     2     5     6  
      -----------------------------------------------------------------------
               2                                   2
      x x  - 3x  + 9x x  + 9x x  + x x  + x x  + 6x  + 3x  + 3x  - x  - x  +
       6 7     7     2 8     5 8    6 8    7 8     8     2     5    6    7  
      -----------------------------------------------------------------------
      5x  + 1
        8

o13 : Ch