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 rational number
- A, a central hyperplane arrangement, 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|  |  | | |  |
     +--+--+--+--+--+--+-+-+--+

Ways to use this method: