BasicDivisor -- the Types of divisors

Description

The BasicDivisor is the class of divisors whose coefficients are unspecified, a base class. Not typically for use. All subtypes have the same essential structure.

RWeilDivisor is a subclass which has real coefficients.

QWeilDivisor is a further subclass with rational coefficients.

WeilDivisor is a subclass with integer coefficients.

The basic structure is a HashTable. There is one key which has a value which specifies the ambient ring. Another key is cache which points to a CacheTable. The remaining keys are a Groebner basis $L$ for each prime ideal $P$ in the support with corresponding value a list with one entry {$n$} where $n$ is the coefficient of the height one prime.

i1 : R = QQ[x,y,z];

i2 : D = divisor(x*y^2*z^3)

o2 = Div(x) + 2*Div(y) + 3*Div(z)

o2 : WeilDivisor on R

i3 : H = new HashTable from D

o3 = HashTable{{x} => {1}                  }
               {y} => {2}
               {z} => {3}
               cache => CacheTable{...1...}
               ring => R

o3 : HashTable

i4 : (2/3)*D

o4 = 2/3*Div(x) + 2*Div(z) + 4/3*Div(y)

o4 : QWeilDivisor on R

i5 : 0.6*D

o5 = .6*Div(x) + 1.8*Div(z) + 1.2*Div(y)

o5 : RWeilDivisor on R

Types of BasicDivisor:

RWeilDivisor

Functions and methods returning an object of class BasicDivisor:

applyToCoefficients(BasicDivisor,Function) -- see applyToCoefficients -- apply a function to the coefficients of a divisor
- BasicDivisor -- see BasicDivisor + BasicDivisor -- add or subtract two divisors, or negate a divisor
BasicDivisor + BasicDivisor -- add or subtract two divisors, or negate a divisor
BasicDivisor - BasicDivisor -- see BasicDivisor + BasicDivisor -- add or subtract two divisors, or negate a divisor
cleanSupport(BasicDivisor) -- see cleanSupport -- removes primes with coefficient zero from a divisor
clearCache(BasicDivisor) -- see clearCache -- creates a new divisor with most entries from the cache removed
divisor(BasicList) -- see divisor -- constructor for (Weil/Q/R)-divisors
divisor(BasicList,BasicList) -- see divisor -- constructor for (Weil/Q/R)-divisors
Number * BasicDivisor -- multiply a divisor by a number
pullback(RingMap,RWeilDivisor) -- pullback a divisor under a ring map
trim(BasicDivisor) -- trims the ideals displayed to the user and removes primes with coefficient zero
zeroDivisor(Ring) -- see zeroDivisor -- constructs the zero Weil divisor for the ring

Methods that use an object of class BasicDivisor:

coefficient(BasicList,BasicDivisor) -- get the coefficient of an ideal for a fixed divisor
coefficient(Ideal,BasicDivisor) -- get the coefficient of an ideal for a fixed divisor
coefficients(BasicDivisor) -- get the list of coefficients of a divisor
gbs(BasicDivisor) -- see gbs -- get the list of Groebner bases corresponding to the height-one primes in the support of a divisor
getPrimeCount(BasicDivisor) -- see getPrimeCount -- get the number of height-one primes in the support of the divisor
getPrimeDivisors(BasicDivisor) -- see getPrimeDivisors -- get the list of prime divisors of a given divisor
isEffective(BasicDivisor) -- see isEffective -- whether a divisor is effective
isHomogeneous(BasicDivisor) -- whether the divisor is graded (homogeneous)
isPrime(BasicDivisor) -- whether a divisor is prime
isReduced(BasicDivisor) -- see isReduced -- whether a divisor is reduced
isSNC(BasicDivisor) -- see isSNC -- whether the divisor is simple normal crossings
isWellDefined(BasicDivisor) -- whether a divisor is valid
isZeroDivisor(BasicDivisor) -- see isZeroDivisor -- whether the divisor is the zero divisor
net(BasicDivisor) (missing documentation)
primes(BasicDivisor) -- see primes -- get the list of height-one primes in the support of a divisor
ring(BasicDivisor) -- get the ambient ring of a divisor

For the programmer

The object BasicDivisor is a type, with ancestor classes HashTable < Thing.

The source of this document is in Divisor.m2:1993:0.