cotangentSheaf(NormalToricVariety) -- make the sheaf of Zariski 1-forms

Synopsis

Function: cotangentSheaf
Usage:

cotangentSheaf X
Inputs:
- X, a normal toric variety,
Optional inputs:
- MinimalGenerators => a Boolean value, default value true, that specifies whether to apply minimalPresentation to the result before returning it
- Strategy (missing documentation) => ..., default value null,
Outputs:
- a coherent sheaf, the sheaf of Zariski 1-forms on X

Description

For a normal variety, the sheaf of Zariski 1-forms is defined to be the double dual of the cotangent bundle or equivalently the extension of the sheaf of 1-forms on the smooth locus to the entire variety (the complement of the smooth locus has codimension at least two because the variety is normal). By construction, this sheaf is reflexive with rank equal to the dimension of the variety. When the underlying variety is smooth, this is simple the sheaf of 1-forms or the cotangent bundle. For more information, see Theorem 8.1.5 in Cox-Little-Schenck's Toric Varieties.

On a non-degenerate normal toric variety, the sheaf of Zariski 1-forms is associated to the kernel of a map from the character lattice tensor the total coordinate ring to the direct sum over the rays of the quotient of the total coordinate ring by the ideal generated by the corresponding variable.

i1 : PP3 = toricProjectiveSpace 3;

i2 : OmegaPP3 = cotangentSheaf PP3

o2 = cokernel {2} | x_2  0    0    x_3  |
              {2} | x_0  x_3  0    0    |
              {2} | -x_1 0    x_3  0    |
              {2} | 0    x_1  x_0  0    |
              {2} | 0    -x_2 0    x_0  |
              {2} | 0    0    -x_2 -x_1 |

                                             6
o2 : coherent sheaf on PP3, quotient of OO    (-2)
                                          PP3

i3 : assert (prune cotangentSheaf PP3 === cotangentSheaf (PP3, MinimalGenerators => true))

i4 : L = prune exteriorPower (3, OmegaPP3)

          1
o4 = OO    (-4)
       PP3

o4 : coherent sheaf on PP3, free of rank 1

i5 : assert (L === OO toricDivisor PP3)

i6 : assert (L === prune cotangentSheaf (dim PP3, PP3))

i7 : X = hirzebruchSurface 2;

i8 : OmegaX = cotangentSheaf X

o8 = cokernel {2, 0}  | 2x_1x_3 |
              {-1, 2} | x_0     |
              {-1, 2} | -x_2    |

                                         1               2
o8 : coherent sheaf on X, quotient of OO   (-2, 0) ++ OO   (1, -2)
                                        X               X

i9 : L = prune exteriorPower(dim X, OmegaX)

        1
o9 = OO   (0, -2)
       X

o9 : coherent sheaf on X, free of rank 1

i10 : assert (L === OO toricDivisor X)

i11 : assert (L === prune cotangentSheaf(dim X, X))

i12 : Y = normalToricVariety ({{1,0,0},{0,1,0},{0,0,1},{0,-1,-1},{-1,0,-1},{-2,-1,0}}, {{0,1,2},{0,1,3},{1,3,4},{1,2,4},{2,4,5},{0,2,5},{0,3,5},{3,4,5}});

i13 : assert (not isSmooth Y and not isProjective Y)

i14 : OmegaY = cotangentSheaf Y

o14 = cokernel {2, 0, 2} | x_2x_4  2x_0x_4 0       0    0       0     3x_0x_2 0     0       |
               {2, 2, 0} | -x_1x_5 0       x_0x_5  0    0       0     0       0     3x_0x_1 |
               {0, 2, 3} | 0       -x_1x_3 -x_2x_3 0    3x_1x_2 0     0       0     0       |
               {2, 2, 2} | x_3     0       0       3x_0 0       0     0       0     0       |
               {2, 2, 3} | 0       -x_5    0       0    0       0     0       3x_2  0       |
               {2, 2, 3} | 0       0       -x_4    0    0       3x_1  0       0     0       |
               {2, 2, 4} | 0       0       0       -x_4 2x_4    -2x_3 x_3     0     0       |
               {2, 3, 3} | 0       0       0       -x_5 -x_5    0     0       x_3   -x_3    |
               {3, 2, 3} | 0       0       0       0    0       -x_5  -x_5    -2x_4 -x_4    |

                                          1                   1                   1                   1                    2                    1                    1                    1
o14 : coherent sheaf on Y, quotient of OO   (-2, 0, -2) ++ OO   (-2, -2, 0) ++ OO   (0, -2, -3) ++ OO   (-2, -2, -2) ++ OO   (-2, -2, -3) ++ OO   (-2, -2, -4) ++ OO   (-2, -3, -3) ++ OO   (-3, -2, -3)
                                         Y                   Y                   Y                   Y                    Y                    Y                    Y                    Y

i15 : prune exteriorPower(dim Y, OmegaY)

         1
o15 = OO   (-3, -3, -4)
        Y

o15 : coherent sheaf on Y, free of rank 1

i16 : assert (prune exteriorPower(dim Y, OmegaY) === OO toricDivisor Y)

Ways to use this method: