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NormalToricVarieties :: abstractSheaf(NormalToricVariety,AbstractVariety,ToricDivisor)

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

Synopsis

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 , t , t ]
                       0   1   2   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 , t , t ]
                       0   1   2   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 = normalToricVariety((({{1, 0}, {0, 1}, {-1, 2}, {0, -1}}(,({{0, 1}, {0, 3}, {1, 2}, {2, 3}} )))))

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 , t , t ]
                    0   1   2   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)

See also