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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

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