# orlikTerao -- defining ideal for the Orlik-Terao algebra

## Synopsis

• Usage:
orlikTerao(A) or orlikTerao(A,S) or orlikTerao(A,y)
• Inputs:
• A, , a central hyperplane arrangement
• S, a ring, a commutative polynomial ring with one variable for each hyperplane
• y, , a name for an indexed variable
• Optional inputs:
• NaiveAlgorithm (missing documentation) => ..., default value false,
• Outputs:
• an ideal, the defining ideal of the Orlik-Terao algebra of A

## 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 :

• "orlikTerao(CentralArrangement)"
• "orlikTerao(CentralArrangement,PolynomialRing)"
• "orlikTerao(CentralArrangement,Symbol)"

## For the programmer

The object orlikTerao is .