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# deletion(Arrangement,RingElement) -- deletion of a subset of an arrangement

## Synopsis

• Function: deletion
• Usage:
deletion(A,x)
deletion(A,S)
deletion(A,i)
• Inputs:
• x, , alternatively, the second argument can be the index of a hyperplane, or a set or list of indices of hyperplanes
• Outputs:
• , obtained by deleting the linear form $x$, or the subset $S$, or the $i$th linear form

## Description

The deletion is obtained by removing hyperplanes from ${\mathcal A}$.

 i1 : A = arrangement "braid" o1 = {x , x , x , x - x , x - x , x - x } 1 2 3 1 2 1 3 2 3 o1 : Hyperplane Arrangement  i2 : deletion(A,5) o2 = {x , x , x , x - x , x - x } 1 2 3 1 2 1 3 o2 : Hyperplane Arrangement 

You can also remove a hyperplane by specifying its linear form.

 i3 : R = QQ[x,y] o3 = R o3 : PolynomialRing i4 : A = arrangement {x,y,x-y} o4 = {x, y, x - y} o4 : Hyperplane Arrangement  i5 : deletion(A, x-y) o5 = {x, y} o5 : Hyperplane Arrangement 

If multiple linear forms define the same hyperplane $H$, deleting any one of those forms does the same thing: it finds the first linear form in $\mathcal A$ defining $H$, then deletes that one.

 i6 : A = arrangement {x, x-y, y, x-y, y-x} o6 = {x, x - y, y, x - y, - x + y} o6 : Hyperplane Arrangement  i7 : A1 = deletion(A, x-y) o7 = {x, y, x - y, - x + y} o7 : Hyperplane Arrangement  i8 : A2 = deletion(A, y-x) o8 = {x, y, x - y, - x + y} o8 : Hyperplane Arrangement  i9 : A3 = deletion(A, 2*(x-y)) o9 = {x, y, x - y, - x + y} o9 : Hyperplane Arrangement  i10 : assert(A1 == A2) i11 : assert(A2 == A3)