### D.15.5 divisors_lib

Library:
  divisors.lib
Purpose:
  Divisors and P-Divisors

Authors:
  Janko Boehm boehm@mathematik.uni-kl.de
  Lars Kastner kastner@math.fu-berlin.de
  Benjamin Lorenz blorenz@math.uni-frankfurt.de
  Hans Schoenemann hannes@mathematik.uni-kl.de
  Yue Ren ren@mathematik.uni-kl.de

Overview:
  We implement a class divisor on an algebraic variety and methods for computing with them. Divisors are represented by tuples of ideals defining the positive and the negative part. In particular, we implement the group structure on divisors, computing global sections and testing linear equivalence.

  In addition to this we provide a class formaldivisor which implements integer formal sums of divisors (not necessarily prime). A formal divisor can be evaluated to a divisor, and a divisor can be decomposed into a formal sum.

  Finally we provide a class pdivisor which implements polyhedral formal sums of divisors (P-divisors) where the coefficients are assumed to be polyhedra with fixed tail cone. There is a function to evaluate a P-divisor on a vector in the dual of the tail cone. The result will be a formal divisor.

References:
  For the class divisor we closely follow Macaulay2's tutorial on divisors.

Procedures:

> D.15.5.1 makeDivisor |  | create a divisor
> D.15.5.2 divisorplus |  | add two divisors
> D.15.5.3 multdivisor |  | multiply a divisor by an interger
> D.15.5.4 negativedivisor |  | compute the negative of the divisor
> D.15.5.5 normalForm |  | normal form of a divisor
> D.15.5.6 isEqualDivisor |  | test whether two divisors are equal
> D.15.5.7 globalSections |  | compute the global sections of a divisor
> D.15.5.8 degreeDivisor |  | degree of a divisor
> D.15.5.9 linearlyEquivalent |  | test whether two divisors a linearly equivalent
> D.15.5.10 effective |  | compute an effective divisor linearly equivalent to a divisor
> D.15.5.11 makeFormalDivisor |  | make a formal integer sum of divisors
> D.15.5.12 evaluateFormalDivisor |  | evalutate a formal sum of divisors to a divisor
> D.15.5.13 formaldivisorplus |  | add two formal divisors
> D.15.5.14 negativeformaldivisor |  | compute the negative of the formal divisor
> D.15.5.15 multformaldivisor |  | multiply a formal divisor by an interger
> D.15.5.16 degreeFormalDivisor |  | degree of a formal divisor
> D.15.5.17 makePDivisor |  | make a formal polyhedral sum of divisors
> D.15.5.18 evaluatePDivisor |  | evaluate a polyhedral divisor to an integer formal divisor
> D.15.5.19 pdivisorplus |  | add two polyhedral divisors

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