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];
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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
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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
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i4 : betti res I
0 1 2 3
o4 = total: 1 4 5 2
0: 1 . . .
1: . 4 2 .
2: . . 3 2
o4 : BettiTally
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i5 : OT := comodule I;
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i6 : apply(1+dim OT, i -> 0 == Ext^i(OT, ring OT))
o6 = {true, true, true, false}
o6 : List
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The Orlik-Terao algebra is always Cohen-Macaulay (Proudfoot-Speyer, 2006).