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# globalSections -- The global sections of a toric divisor.

- Usage:
    `globalSections(A,v)`
    `globalSections(A,v,L)`
- Inputs:
  - A, a matrix,
  - v, a vector,
  - L, a list,
- Outputs:
  - a list,

## Description

Computes the global sections of a toric Weil divisor D with coefficients v with respect to the coker grading by A. In the same way as v they are represented by vectors (exponent vectors of Laurent monomials in rank target A variables).

If a list of indices L in {0..rank target A -1} is specified, then those Laurent monomial exponents are computed, which induce a linear equivalence of D to an effective divisor with support precisely on L.

    i1 : A=matrix {{1, 0}, {0, 1}, {-1, -1}}

    o1 = | 1  0  |
         | 0  1  |
         | -1 -1 |

                  3       2
    o1 : Matrix ZZ  <-- ZZ

    i2 : b=vector {2,0,0}

    o2 = | 2 |
         | 0 |
         | 0 |

           3
    o2 : ZZ

    i3 : globalSections(A,b)

    o3 = {| -2 |, | -2 |, | -2 |, | -1 |, | -1 |, 0}
          |  0 |  |  1 |  |  2 |  |  0 |  |  1 |
          |  2 |  |  1 |  |  0 |  |  1 |  |  0 |

    o3 : List

    i4 : A=matrix {{1, 0}, {0, 1}, {-1, -1},{1,1}}

    o4 = | 1  0  |
         | 0  1  |
         | -1 -1 |
         | 1  1  |

                  4       2
    o4 : Matrix ZZ  <-- ZZ

    i5 : b=vector {2,0,0,0}

    o5 = | 2 |
         | 0 |
         | 0 |
         | 0 |

           4
    o5 : ZZ

    i6 : globalSections(A,b)

    o6 = {| -2 |, | -1 |, 0}
          |  2 |  |  1 |
          |  0 |  |  0 |
          |  0 |  |  0 |

    o6 : List

    i7 : globalSections(A,b,{1})

    o7 = {| -2 |}
          |  2 |
          |  0 |
          |  0 |

    o7 : List

## Caveat

This uses the package Polyhedra (if ConvexInterface.m2 is not present) to compute the lattice points of a convex hull.

## Ways to use globalSections:

- `globalSections(Matrix,Vector)`
- `globalSections(Matrix,Vector,List)`

## For the programmer

The object globalSections is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SRdeformations.m2:6407:0.