MultiProjCoordRing -- A quick way to build the coordinate ring of a product of projective spaces

Synopsis

Usage:

MultiProjCoordRing Dims

MultiProjCoordRing (CoeffRing,Dims)

MultiProjCoordRing (var,Dims)

MultiProjCoordRing (CoeffRing,var,Dims)
Inputs:
- Dims, a list, representing the dimensions of the projective spaces, i.e. {n_1,...,n_m} corresponds to \PP^{n_1} x.... x \PP^{n_m}
- CoeffRing, a ring, the coefficient ring of the graded polynomial ring to be built by the method, by default this is \ZZ/32749
- var, a symbol, to be used for the intermediates of the graded polynomial ring to be built by the method
Outputs:
- a ring, the graded coordinate ring of the \PP^{n_1} x.... x \PP^{n_m} where {n_1,...,n_m} is the input list of dimensions

Description

Computes the graded coordinate ring of the \PP^{n_1} x.... x \PP^{n_m} where {n_1,...,n_m} is the input list of dimensions. This method is used to quickly build the coordinate ring of a product of projective spaces for use in computations.

i1 : S=MultiProjCoordRing(QQ,symbol z,{1,3,3})

o1 = S

o1 : PolynomialRing

i2 : degrees S

o2 = {{1, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0,
     ------------------------------------------------------------------------
     0, 1}, {0, 0, 1}, {0, 0, 1}, {0, 0, 1}}

o2 : List

i3 : R=MultiProjCoordRing {2,3}

o3 = R

o3 : PolynomialRing

i4 : coefficientRing R

       ZZ
o4 = -----
     32749

o4 : QuotientRing

i5 : describe R

       ZZ
o5 = -----[x ..x , Degrees => {3:{1}, 4:{0}}, Heft => {2:1}]
     32749  0   6                {0}    {1}

i6 : A=ChowRing R

o6 = A

o6 : QuotientRing

i7 : describe A

     ZZ[h ..h ]
         1   2
o7 = ----------
        3   4
      (h , h )
        1   2

i8 : Segre(A,ideal random({1,1},R))

        2 3     2 2       3     2         2    3    2            2
o8 = 10h h  - 6h h  - 4h h  + 3h h  + 3h h  + h  - h  - 2h h  - h  + h  + h
        1 2     1 2     1 2     1 2     1 2    2    1     1 2    2    1    2

o8 : A

Ways to use MultiProjCoordRing:

MultiProjCoordRing(List)
MultiProjCoordRing(Ring,List)
MultiProjCoordRing(Ring,Symbol,List)
MultiProjCoordRing(Symbol,List)

For the programmer

The object MultiProjCoordRing is a method function.