idealSheaf -- ideal sheaf of a projective variety

Usage:

idealSheaf X
Inputs:
- X, a projective variety,
Optional inputs:
- MinimalGenerators (missing documentation) => a Boolean value, default value true, whether to prune the result before returning it. Default value is true.
- Strategy (missing documentation) => ..., default value null,
Outputs:
- a coherent sheaf,

Description

This method computes the ideal sheaf of the projective variety $X$

As an example, consider the projective variety defined by the equation $x^4 + y^4 + z^4 = 0$. The ideal sheaf of this variety is computed.

i1 : X = Proj QQ[x,y,z]/(x^4+y^4+z^4)

o1 = X

o1 : ProjectiveVariety

i2 : I = idealSheaf X

o2 = image | x4+y4+z4 |

                                                                   1
o2 : coherent sheaf on Proj(QQ[x..z]), subsheaf of OO
                                                     Proj(QQ[x..z])

i3 : rho = inducedMap((ambient I)/I, ambient I) --induced inclusion of ideal sheaf into structure sheaf of ambient ring

                                                      1
o3 = cokernel | x4+y4+z4 | <--------- OO
                              | 1 |     Proj(QQ[x..z])

o3 : SheafMap

i4 : for i to 2 list HH^i(image rho)

        1    3
o4 = {QQ , QQ , 0}

o4 : List

Ways to use idealSheaf:

idealSheaf(ProjectiveVariety)

For the programmer

The object idealSheaf is a method function with options.

The source of this document is in Varieties/doc-sheaves.m2:501:0.

idealSheaf -- ideal sheaf of a projective variety

Description

See also

Ways to use idealSheaf:

For the programmer