Singular

D.5.4 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.5.4.1 makeDivisor		create a divisor
D.5.4.2 divisorplus		add two divisors
D.5.4.3 multdivisor		multiply a divisor by an integer
D.5.4.4 negativedivisor		compute the negative of the divisor
D.5.4.5 normalForm		normal form of a divisor
D.5.4.6 isEqualDivisor		test whether two divisors are equal
D.5.4.7 globalSections		compute the global sections of a divisor
D.5.4.8 degreeDivisor		degree of a divisor
D.5.4.9 linearlyEquivalent		test whether two divisors a linearly equivalent
D.5.4.10 effective		compute an effective divisor linearly equivalent to a divisor
D.5.4.11 makeFormalDivisor		make a formal integer sum of divisors
D.5.4.12 evaluateFormalDivisor		evaluate a formal sum of divisors to a divisor
D.5.4.13 formaldivisorplus		add two formal divisors
D.5.4.14 negativeformaldivisor		compute the negative of the formal divisor
D.5.4.15 multformaldivisor		multiply a formal divisor by an integer
D.5.4.16 degreeFormalDivisor		degree of a formal divisor
D.5.4.17 makePDivisor		make a formal polyhedral sum of divisors
D.5.4.18 evaluatePDivisor		evaluate a polyhedral divisor to an integer formal divisor
D.5.4.19 pdivisorplus		add two polyhedral divisors