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HyperplaneArrangements :: orlikSolomon

orlikSolomon -- defining ideal for the Orlik-Solomon algebra

Synopsis

Description

The Orlik-Solomon algebra is the cohomology ring of the complement of the hyperplanes, either in complex projective or affine space. The optional Boolean argument Projective specifies which. The code for this method was written by Sorin Popescu.
i1 : A = typeA(3)

o1 = {x  - x , x  - x , x  - x , x  - x , x  - x , x  - x }
       1    2   1    3   1    4   2    3   2    4   3    4

o1 : Hyperplane Arrangement 
i2 : I = orlikSolomon(A,e)

o2 = ideal (e e  - e e  + e e , e e  - e e  + e e , e e  - e e  + e e , e e 
             4 5    4 6    5 6   2 3    2 6    3 6   1 3    1 5    3 5   1 2
     ------------------------------------------------------------------------
     - e e  + e e )
        1 4    2 4

o2 : Ideal of QQ[e ..e ]
                  1   6
i3 : reduceHilbert hilbertSeries I

                 2     3
     1 + 6T + 11T  + 6T
o3 = -------------------
              1

o3 : Expression of class Divide
i4 : I' = orlikSolomon(A,Projective=>true,HypAtInfinity=>2)

o4 = ideal (e e  - e e  + e e , e e  - e e  + e e , e e  - e e  + e e , e e 
             4 5    4 6    5 6   2 3    2 6    3 6   1 3    1 5    3 5   1 2
     ------------------------------------------------------------------------
     - e e  + e e , e )
        1 4    2 4   3

o4 : Ideal of QQ[e ..e ]
                  1   6
i5 : reduceHilbert hilbertSeries I'

                2
     1 + 5T + 6T
o5 = ------------
           1

o5 : Expression of class Divide

The code for orlikSolomon was contributed by Sorin Popescu.

Ways to use orlikSolomon :

For the programmer

The object orlikSolomon is a method function with options.