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HyperplaneArrangements :: orlikTerao

orlikTerao -- defining ideal for the Orlik-Terao algebra

Synopsis

Description

The Orlik-Terao algebra of an arrangement is the subalgebra of rational functions k[1/f_1,1/f_2,...,1/f_n] where the f_i's are the defining forms for the hyperplanes. This method produces the kernel of the obvious surjection from a polynomial ring in n variables onto the Orlik-Terao algebra.
i1 : R := QQ[x,y,z];
i2 : orlikTerao arrangement {x,y,z,x+y+z}

o2 = ideal(y y y  - y y y  - y y y  - y y y )
            1 2 3    1 2 4    1 3 4    2 3 4

o2 : Ideal of QQ[y ..y ]
                  1   4
The defining ideal above has one generator given by the single relation x+y+z-(x+y+z)=0. The rank-3 braid arrangement has four triple points:
i3 : I := orlikTerao arrangement "braid"

o3 = ideal (y y  - y y  + y y , y y  + y y  - y y , y y  + y y  - y y , y y 
             4 5    4 6    5 6   2 3    2 6    3 6   1 3    1 5    3 5   1 2
     ------------------------------------------------------------------------
     + y y  - y y )
        1 4    2 4

o3 : Ideal of QQ[y ..y ]
                  1   6
i4 : betti res I

            0 1 2 3
o4 = total: 1 4 5 2
         0: 1 . . .
         1: . 4 2 .
         2: . . 3 2

o4 : BettiTally
i5 : OT := comodule I;
i6 : apply(1+dim OT, i -> 0 == Ext^i(OT, ring OT))	  

o6 = {true, true, true, false}

o6 : List
The Orlik-Terao algebra is always Cohen-Macaulay (Proudfoot-Speyer, 2006).

Ways to use orlikTerao :

For the programmer

The object orlikTerao is a method function with options.