globalSections -- The global sections of a toric divisor.

Synopsis

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 OldPolyhedra.m2 (if ConvexInterface.m2 is not present) to compute the lattice points of a convex hull. constructHilbertBasis of the package OldPolyhedra.m2 used by latticePoints overwrites global variable C. Fixed this in my local version.

Ways to use globalSections :

globalSections(Matrix,Vector)
globalSections(Matrix,Vector,List)

For the programmer

The object globalSections is a method function.