smoothDivisors -- produces a list of smooth divisors of a given degree on a surface

Description

Produces a List of Divisor of a given degree on a given surface. On the CubicSurface and the QuarticSurfaceRational the list is not exhaustive, but will include a divisor for each possible genus that a smooth curve can obtain.

Synopsis

Usage:

L = smoothDiviors(d,X)
Inputs:
- d, an integer, degree
- X, an instance of the type QuadricSurface
Outputs:
- L, a list, of Divisors

i1 : X = quadricSurface(ZZ/101[x,y,z,w]);

i2 : L = smoothDivisors(5,X)

o2 = {{1, 4}, {2, 3}}

o2 : List

Synopsis

Usage:

L = smoothDiviors(d,X)
Inputs:
- d, an integer, degree
- X, an instance of the type CubicSurface
Outputs:
- L, a list, of Divisors

i3 : X = cubicSurface(ZZ/101[x,y,z,w]);

i4 : L = smoothDivisors(5,X)

o4 = {{2, 1, 0, 0, 0, 0, 0}, {3, 1, 1, 1, 1, 0, 0}, {4, 2, 1, 1, 1, 1, 1}}

o4 : List

Synopsis

Usage:

L = smoothDiviors(d,X)
Inputs:
- d, an integer, degree
- X, an instance of the type QuarticSurfaceRational
Outputs:
- L, a list, of Divisors

i5 : X = quarticSurfaceRational(ZZ/101[x,y,z,w]);

i6 : L = smoothDivisors(5,X)

o6 = {{3, 0, 2, 1, 1, 1, 1, 1, 0, 0}, {3, 0, 1, 1, 1, 1, 1, 1, 1, 0}, {4, 1,
     ------------------------------------------------------------------------
     2, 1, 1, 1, 1, 1, 1, 1}}

o6 : List

Ways to use smoothDivisors :

smoothDivisors(ZZ,CubicSurface)
smoothDivisors(ZZ,QuadricSurface)
smoothDivisors(ZZ,QuarticSurfaceRational)
smoothDivisors(ZZ,ZZ) -- produces a list of smooth divisors of given degree and genus
smoothDivisors(ZZ,ZZ,Ring) -- see smoothDivisors(ZZ,ZZ) -- produces a list of smooth divisors of given degree and genus

For the programmer

The object smoothDivisors is a method function.