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deCone(CentralArrangement,RingElement) -- produce an affine arrangement from a central one

Synopsis

• Function: deCone
• Usage:
deCone(A, x)
deCone(A, i)
• Inputs:
• x, , a hyperplane of $A$ or the index of a hyperplane of $A$
• Outputs:
• , the decone of $A$ over $x$

Description

The decone of a central arrangement $A$ at a hyperplane $H=H_i$ or $H=\ker x$ is the affine arrangement obtained from $A$ by first deleting the hyperplane $H$ then intersecting the remaining hyperplanes with the (affine) hyperplane $\{x=1\}$. In particular, if $R$ is the coordinate ring of $A$, then the coordinate ring of its decone over $x$ is $R/(x-1)$.

The decone of a central arrangement at $H$ can also be constructed by first projectivizing $A$, then removing the image of $H$, and identifying the complement of $H$ with affine space.

 i1 : A = arrangement "X3" o1 = {x , x , x , x + x , x + x , x + x } 1 2 3 1 2 1 3 2 3 o1 : Hyperplane Arrangement  i2 : dA = deCone(A,2) o2 = {x , x , x + x , x + 1, x + 1} 1 2 1 2 1 2 o2 : Hyperplane Arrangement  i3 : factor poincare A 2 o3 = (1 + T)(1 + 5T + 7T ) o3 : Expression of class Product i4 : poincare dA 2 o4 = 1 + 5T + 7T o4 : ZZ[T]

The coordinate ring of $dA$ is $\mathbb{Q}[x_1,x_2,x_3]/(x_3-1)$.

 i5 : ring dA QQ[x ..x ] 1 3 o5 = ---------- x - 1 3 o5 : QuotientRing

Use prune to get something whose coordinate ring is a polynomial ring.

 i6 : dA' = prune dA o6 = {x , x , x + x , x + 1, x + 1} 1 2 1 2 1 2 o6 : Hyperplane Arrangement  i7 : ring dA' o7 = QQ[x ..x ] 1 2 o7 : PolynomialRing