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HyperplaneArrangements :: multIdeal

multIdeal -- compute a multiplier ideal

Synopsis

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];
i2 : A := arrangement ((x^2 - y^2)*(x^2 - z^2)*(y^2 - z^2)*z);
i3 : multIdeal(3/7,A)

o3 = ideal (z, y, x)

o3 : Ideal of QQ[x..z]
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 := multIdeal(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|  |  | | |  |
     +--+--+--+--+--+--+-+-+--+

See also

Ways to use multIdeal :

For the programmer

The object multIdeal is a method function.