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Divisor
::
Divisor
Divisor -- A package for divisors on normal rings (graded or not).
Description
A package for handling Weil divisors on normal rings, graded or not.
Authors
Karl Schwede
<
kschwede@gmail.com
>
Zhaoning Yang
<
zyy5054@gmail.com
>
Version
This documentation describes version
0.1p
of Divisor.
Source code
The source code from which this documentation is derived is in the file
Divisor.m2
.
Exports
Types
BasicDiv
-- the class of divisors with unspecified coefficients
QDiv
-- the class of divisors with rational coefficients
RDiv
-- the class of divisors with real coefficients
WDiv
-- the class of divisors with integer coefficients
Functions and commands
applyToCoefficients
-- Applies a function to the coefficients of a divisor
baseLocus
-- Computes the locus where a graded module (or O(D) Weil divisor) is not globally generated.
canonicalDivisor
-- Compute the canonical divisor of a ring
ceilingDiv
-- Get a divisor whose coefficients are ceilings of the given divisor
coeff
-- Get the coefficient of a given ideal for a fixed divisor
divisor
-- Constructor for (Weil/Q/R)-divisors
divisorToIdeal
-- Calculate the corresponding module of a given divisor and represent it as an ideal
divisorToModule
-- Calculate the corresponding module of a given divisor
divMinus
-- Get the negative part of a divisor
divPlus
-- Get the positive part of a given divisor
divPullBack
-- Compute the pullback of a divisor under a ring map
dualizeIdeal
-- Finds an ideal isomorphic to Hom(I, R)
findElementOfDegree
-- Find an element of a specified degree
floorDiv
-- Get a divisor whose coefficients are floors of the given divisor
getAmbientRing
-- Get the ambient ring of a divisor
getCoeffList
-- Get the list of coefficients of a divisor
getGBList
-- Get the list of Groebner bases corresponding to the height-one primes in the support of a divisor
getLinearDiophantineSolution
-- Find a solution of the linear Diophantine equation Ax = b
getPrimeCount
-- Get the number of height one primes in the support of the divisor
getPrimeDivisors
-- Returns the list of prime divisors of a given divisor
getPrimeList
-- Get the list of height-one primes in the support of a divisor
idealPower
-- Compute the ideal generated by the generators of the given ideal raised to a power
idealToDivisor
-- Calculate the divisor D so that O_X(D) = I
idealWithSectionToDivisor
-- Calculate the divisor D so that D corresponds to the section f of I
isCartier
-- Check if a Weil divisor is Cartier
isDivAmbient
-- Checks whether the ambient ring of a given divisor is the given ring
isDivGraded
-- Checks to see if the divisor is graded (homogeneous)
isDivPrime
-- Check if a divisor is prime
isDivPrincipal
-- Check if a Weil divisor is globally principal
isDivReduced
-- Check if a divisor is reduced
isDomain
-- Checks if a ring is a domain
isEffective
-- Check if a divisor is effective
isLinearEquivalent
-- Check if two Weil divisor are linearly equivalent
isQCartier
-- Check whether m times a divisor is Cartier for any m from 1 to a fixed positive integer n1.
isQLinearEquivalent
-- Check if two rational divisors are linearly equivalent
isReflexive
-- Checks whether an ideal or module is reflexive
isRegular
-- Checks to see if R mod the given ideal is regular
isSNC
-- Checks to see if the divisor is simple normal crossings
isWDiv
-- Check if a rational/real divisor is a Weil divisor
isZeroDivisor
-- Checks to see if the divisor is the zero divisor
mapToProjectiveSpace
-- Compute the map to projective space associated with the global sections of a Cartier divisor
moduleToDivisor
-- Compute a divisor associated to a module in a ring
moduleToIdeal
-- Turn a module to an ideal of a ring
moduleWithSectionToDivisor
-- Compute the effective divisor associated to the section of a module
moduleWithSectionToIdeal
-- Turn a module to an ideal of a ring and keep track of a module element
nonCartierLocus
-- Returns the non-Cartier locus of a Weil divisor
ramificationDivisor
-- Compute the ramification divisor of a finite inclusion of normal domains
rationalDivisor
-- Constructs a Q-divisor
realDivisor
-- Constructs an R-divisor
reflexifyIdeal
-- Calculate the double dual of an ideal
reflexifyModule
-- Calculate the double dual of a module
reflexifyModuleWithMap
-- Compute the canonical map from a module to its double-dual
reflexivePower
-- Computes a reflexive power of an ideal
sameDivAmbient
-- Checks whether the ambient ring of the given divisors are equal
simplifyDiv
-- Removes primes with coefficient zero
toQDiv
-- Turn a Weil divisor to a rational divisor
toRDiv
-- Turn a integer/rational divisor to a real divisor
torsionSubmodule
-- Finds the torsion submodule of a given module
toWDiv
-- Turn a rational/real divisor with integer coefficients into to a Weil Divisor
verifyDivisor
-- Checks to make sure a divisor is valid
zeroDivisor
-- Constructs the zero Weil divisor for the given ring
Methods
- BasicDiv
-- Negation of a divisor
applyToCoefficients(BasicDiv,Function), see
applyToCoefficients
-- Applies a function to the coefficients of a divisor
baseLocus(WDiv), see
baseLocus
-- Computes the locus where a graded module (or O(D) Weil divisor) is not globally generated.
BasicDiv + BasicDiv
-- Sum two divisors.
BasicDiv - BasicDiv
-- Subtract two divisors.
ceilingDiv(RDiv), see
ceilingDiv
-- Get a divisor whose coefficients are ceilings of the given divisor
coeff(BasicList,BasicDiv), see
coeff
-- Get the coefficient of a given ideal for a fixed divisor
coeff(Ideal,BasicDiv), see
coeff
-- Get the coefficient of a given ideal for a fixed divisor
divisorToIdeal(QDiv), see
divisorToIdeal
-- Calculate the corresponding module of a given divisor and represent it as an ideal
divisorToIdeal(RDiv), see
divisorToIdeal
-- Calculate the corresponding module of a given divisor and represent it as an ideal
divisorToIdeal(WDiv), see
divisorToIdeal
-- Calculate the corresponding module of a given divisor and represent it as an ideal
divisorToModule(QDiv), see
divisorToModule
-- Calculate the corresponding module of a given divisor
divisorToModule(RDiv), see
divisorToModule
-- Calculate the corresponding module of a given divisor
divisorToModule(WDiv), see
divisorToModule
-- Calculate the corresponding module of a given divisor
divMinus(RDiv), see
divMinus
-- Get the negative part of a divisor
divPlus(RDiv), see
divPlus
-- Get the positive part of a given divisor
divPullBack(RingMap,RDiv), see
divPullBack
-- Compute the pullback of a divisor under a ring map
floorDiv(RDiv), see
floorDiv
-- Get a divisor whose coefficients are floors of the given divisor
getAmbientRing(BasicDiv), see
getAmbientRing
-- Get the ambient ring of a divisor
getCoeffList(BasicDiv), see
getCoeffList
-- Get the list of coefficients of a divisor
getGBList(BasicDiv), see
getGBList
-- Get the list of Groebner bases corresponding to the height-one primes in the support of a divisor
getPrimeCount(BasicDiv), see
getPrimeCount
-- Get the number of height one primes in the support of the divisor
getPrimeDivisors(BasicDiv), see
getPrimeDivisors
-- Returns the list of prime divisors of a given divisor
getPrimeList(BasicDiv), see
getPrimeList
-- Get the list of height-one primes in the support of a divisor
isCartier(WDiv), see
isCartier
-- Check if a Weil divisor is Cartier
isDivAmbient(BasicDiv,Ring), see
isDivAmbient
-- Checks whether the ambient ring of a given divisor is the given ring
isDivGraded(BasicDiv), see
isDivGraded
-- Checks to see if the divisor is graded (homogeneous)
isDivPrime(BasicDiv), see
isDivPrime
-- Check if a divisor is prime
isDivPrincipal(WDiv), see
isDivPrincipal
-- Check if a Weil divisor is globally principal
isDivReduced(BasicDiv), see
isDivReduced
-- Check if a divisor is reduced
isEffective(BasicDiv), see
isEffective
-- Check if a divisor is effective
isLinearEquivalent(WDiv,WDiv), see
isLinearEquivalent
-- Check if two Weil divisor are linearly equivalent
isQCartier(ZZ,QDiv), see
isQCartier
-- Check whether m times a divisor is Cartier for any m from 1 to a fixed positive integer n1.
isQCartier(ZZ,WDiv), see
isQCartier
-- Check whether m times a divisor is Cartier for any m from 1 to a fixed positive integer n1.
isQLinearEquivalent(QDiv,QDiv), see
isQLinearEquivalent
-- Check if two rational divisors are linearly equivalent
isSNC(BasicDiv), see
isSNC
-- Checks to see if the divisor is simple normal crossings
isWDiv(RDiv), see
isWDiv
-- Check if a rational/real divisor is a Weil divisor
isZeroDivisor(BasicDiv), see
isZeroDivisor
-- Checks to see if the divisor is the zero divisor
mapToProjectiveSpace(WDiv), see
mapToProjectiveSpace
-- Compute the map to projective space associated with the global sections of a Cartier divisor
net(BasicDiv)
-- Controls how divisors are displayed to the user
nonCartierLocus(WDiv), see
nonCartierLocus
-- Returns the non-Cartier locus of a Weil divisor
QQ * RDiv
-- Multiply a real divisor by a rational number
QQ * WDiv
-- Multiply a Weil divisor by a rational number
RDiv == RDiv
-- Check if two divisors are equal
RR * QDiv
-- Multiply a rational divisor by a real number
RR * RDiv
-- Multiply a real divisor by a real number
sameDivAmbient(BasicDiv,BasicDiv), see
sameDivAmbient
-- Checks whether the ambient ring of the given divisors are equal
simplifyDiv(BasicDiv), see
simplifyDiv
-- Removes primes with coefficient zero
toQDiv(QDiv), see
toQDiv
-- Turn a Weil divisor to a rational divisor
toQDiv(WDiv), see
toQDiv
-- Turn a Weil divisor to a rational divisor
toRDiv(QDiv), see
toRDiv
-- Turn a integer/rational divisor to a real divisor
toRDiv(RDiv), see
toRDiv
-- Turn a integer/rational divisor to a real divisor
toRDiv(WDiv), see
toRDiv
-- Turn a integer/rational divisor to a real divisor
toWDiv(RDiv), see
toWDiv
-- Turn a rational/real divisor with integer coefficients into to a Weil Divisor
verifyDivisor(BasicDiv), see
verifyDivisor
-- Checks to make sure a divisor is valid
ZZ * BasicDiv
-- Multiply a divisor by an integer
Symbols
AmbRing
-- An option used to tell divisor construction that a particular ambient ring is expected.
CoeffType
-- An option used to tell divisor construction that a particular type of coefficients are expected.
IsGraded
-- An option used by numerous functions which tells it to treat the divisors as if we were working on the Proj of the ambient ring.
KnownCartier
-- An option used to specify to certain functions that we know that the divisor is Cartier.
KnownNormal
-- An option used to specify to certain functions that we know that the ambient ring is normal.
MTries
-- An option used by moduleToIdeal how many times to try embedding the module as an ideal in a random way.
Primes
-- A value for the option Strategy for the divPullBack method
ReturnMap
-- An option for moduleToIdeal and moduleWithSectionToIdeal
Sheaves
-- A value for the option Strategy for the divPullBack method
Unsafe
-- An option used to tell divisor construction not to do various checks