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], as 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];
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i2 : A := arrangement ((x^2 - y^2)*(x^2 - z^2)*(y^2 - z^2)*z);
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i3 : multIdeal(3/7,A)
o3 = ideal (z, y, x)
o3 : Ideal of QQ[x..z]
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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
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i5 : scan(39,i -> (
s := i/21;
I := multIdeal(s,A);
if not H#?I then H#I = {s} else H#I = H#I|{s}));
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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| | | | | |
+--+--+--+--+--+--+-+-+--+
|