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D.15.11 divisors_lib

Divisors and P-Divisors

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

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.

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


D.15.11.1 makeDivisor  create a divisor
D.15.11.2 divisorplus  add two divisors
D.15.11.3 multdivisor  multiply a divisor by an interger
D.15.11.4 negativedivisor  compute the negative of the divisor
D.15.11.5 normalForm  normal form of a divisor
D.15.11.6 isEqualDivisor  test whether two divisors are equal
D.15.11.7 globalSections  compute the global sections of a divisor
D.15.11.8 degreeDivisor  degree of a divisor
D.15.11.9 linearlyEquivalent  test whether two divisors a linearly equivalent
D.15.11.10 effective  compute an effective divisor linearly equivalent to a divisor
D.15.11.11 makeFormalDivisor  make a formal integer sum of divisors
D.15.11.12 evaluateFormalDivisor  evalutate a formal sum of divisors to a divisor
D.15.11.13 formaldivisorplus  add two formal divisors
D.15.11.14 negativeformaldivisor  compute the negative of the formal divisor
D.15.11.15 multformaldivisor  multiply a formal divisor by an interger
D.15.11.16 degreeFormalDivisor  degree of a formal divisor
D.15.11.17 makePDivisor  make a formal polyhedral sum of divisors
D.15.11.18 evaluatePDivisor  evaluate a polyhedral divisor to an integer formal divisor
D.15.11.19 pdivisorplus  add two polyhedral divisors