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

Usage:

CheckToricVarietyValid X
Inputs:
- X, a normal toric variety, a normal toric variety which is a candidate for an ambient space in which to perform characteristic class computations
Outputs:
- a Boolean value,

Description

Note that if you are working with subvarieties of some product of projective spaces \PP^{n_1}\times \cdots \times \PP^{n_m} 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 ]
                         0   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 perform 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 ]
                                                                                                                0   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

For the programmer

The object CheckToricVarietyValid is a method function.