abstractSheaf(NormalToricVariety,AbstractVariety,ToricDivisor) -- make the corresponding abstract sheaf

Synopsis

Function: abstractSheaf
Usage:

abstractVariety (X, B, D)
Inputs:
- X, a normal toric variety, that is complete and simplicial
- B, an abstract variety, the base variety, when omitted the default base variety of dimension 0 is used
- D, a toric divisor, on X
Optional inputs:
- Rank => ..., default value null, unused
- ChernCharacter => ..., default value null, unused
- ChernClass => ..., default value null, unused
- Name => ..., default value null, unused
Outputs:
- an abstract sheaf,

Description

This method converts a ToricDivisor on a NormalToricVariety into an AbstractSheaf over the corresponding AbstractVariety, as defined in the Schubert2 package.

Since many routines from the Schubert2 package have been overloaded so that they apply directly to toric divisors, this method is primarily of interest to developers.

i1 : PP2 = toricProjectiveSpace 3;

i2 : D1 = abstractSheaf (PP2, PP2_0);

i3 : assert (rank D1 === 1 and variety D1 === abstractVariety PP2)

i4 : chern D1

o4 = 1 + t
          3

                     QQ[][t ..t ]
                           0   3
o4 : -------------------------------------------
     (t t t t , - t  + t , - t  + t , - t  + t )
       0 1 2 3     0    1     0    2     0    3

i5 : ch D1

              1 2   1 3
o5 = 1 + t  + -t  + -t
          3   2 3   6 3

                     QQ[][t ..t ]
                           0   3
o5 : -------------------------------------------
     (t t t t , - t  + t , - t  + t , - t  + t )
       0 1 2 3     0    1     0    2     0    3

i6 : D2 = abstractSheaf (PP2, PP2_1);

i7 : assert (D2 === D1)

i8 : FF2 = hirzebruchSurface 2

o8 = FF2

o8 : NormalToricVariety

i9 : D3 = abstractSheaf (FF2, 2*FF2_2 + FF2_3)

o9 = D3

o9 : an abstract sheaf of rank 1 on a variety

i10 : assert (rank D3 === 1 and variety D3 === abstractVariety FF2)

i11 : chern D3

o11 = 1 + (2t  + t )
             2    3

                  QQ[][t ..t ]
                        0   3
o11 : ------------------------------------
      (t t , t t , t  - t , t  + 2t  - t )
        0 2   1 3   0    2   1     2    3

i12 : D4 = abstractSheaf (FF2, 4*FF2_0 + FF2_1)

o12 = D3

o12 : an abstract sheaf of rank 1 on a variety

i13 : assert (D4 === D3)

Ways to use this method: