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

## Synopsis

• Usage:
globalSections(A,v)
globalSections(A,v,L)
• Inputs:
• A, ,
• v, ,
• L, a list,
• Outputs:

## 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 .