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# multiplierIdeal(QQ,CentralArrangement,List) -- compute a multiplier ideal

## Synopsis

• Function: multiplierIdeal
• Usage:
multiplierIdeal(s,A,m)
multiplierIdeal(s,A)
multIdeal(s,A,m)
multIdeal(s,A)
• Inputs:
• s, , a rational number
• A, , a central hyperplane arrangement
• m, a list, an optional list of positive integer multiplicities
• Outputs:
• an ideal, the multiplier ideal of the arrangement at the value $s$

## Description

The multiplier ideals of an given ideal depend on a nonnegative real parameter. This method computes the multiplier ideals of the defining ideal of a hyperplane arrangement, optionally with multiplicities $m$. This uses the explicit formula of M. Mustata [TAMS 358 (2006), no 11, 5015–5023] simplified by Z. Teitler [PAMS 136 (2008), no 5, 1902–1913].

Let's consider Example 6.3 of Berkesch and Leykin from arXiv:1002.1475v2:

 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : A = arrangement ((x^2 - y^2)*(x^2 - z^2)*(y^2 - z^2)*z) o2 = {z, y - z, y + z, x - z, x + z, x - y, x + y} o2 : Hyperplane Arrangement  i3 : multiplierIdeal(3/7,A) o3 = ideal (z, y, x) o3 : Ideal of R

Since the multiplier ideal is a step function of its real parameter, one tests to see at what values it changes:

 i4 : H = new MutableHashTable o4 = MutableHashTable{} o4 : MutableHashTable i5 : scan(39,i -> ( s := i/21; I := multiplierIdeal(s,A); if not H#?I then H#I = {s} else H#I = H#I|{s})) i6 : netList sort values H -- values of s giving same multiplier ideal +--+--+--+--+--+--+-+-+--+ | | 1| 2|1 | 4| 5|2|1| 8| o6 = |0 |--|--|- |--|--|-|-|--| | |21|21|7 |21|21|7|3|21| +--+--+--+--+--+--+-+-+--+ |3 |10|11| | | | | | | |- |--|--| | | | | | | |7 |21|21| | | | | | | +--+--+--+--+--+--+-+-+--+ |4 |13| | | | | | | | |- |--| | | | | | | | |7 |21| | | | | | | | +--+--+--+--+--+--+-+-+--+ |2 |5 |16|17| | | | | | |- |- |--|--| | | | | | |3 |7 |21|21| | | | | | +--+--+--+--+--+--+-+-+--+ |6 |19|20| | | | | | | |- |--|--| | | | | | | |7 |21|21| | | | | | | +--+--+--+--+--+--+-+-+--+ | |22|23|8 |25|26|9|4|29| |1 |--|--|- |--|--|-|-|--| | |21|21|7 |21|21|7|3|21| +--+--+--+--+--+--+-+-+--+ |10|31|32| | | | | | | |--|--|--| | | | | | | | 7|21|21| | | | | | | +--+--+--+--+--+--+-+-+--+ |11|34| | | | | | | | |--|--| | | | | | | | | 7|21| | | | | | | | +--+--+--+--+--+--+-+-+--+ |5 |12|37|38| | | | | | |- |--|--|--| | | | | | |3 | 7|21|21| | | | | | +--+--+--+--+--+--+-+-+--+