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CharacteristicClasses :: MultiProjCoordRing

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

Synopsis

Description

Computes the graded coordinate ring of the ℙn1 x.... x ℙnm where n1,...,nm 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 , x , x , x , x , x , Degrees => {3:{1}, 4:{0}}, Heft => {2:1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 2]
     32749  0   1   2   3   4   5   6                {0}    {1}                                   {GRevLex => {7:1}  }
                                                                                                  {Position => Up    }
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 :