Description
The log canonical threshold of an ideal I is the infimum of t for which the multiplier ideal J(It) is a proper ideal. Equivalently it is the least nonzero jumping number.
log canonical threshold of a monomial ideal
-
- Usage:
- logCanonicalThreshold I
-
Inputs:
-
Outputs:
Computes the log canonical threshold of a monomial ideal
I.
i1 : R = QQ[x,y];
|
i2 : I = monomialIdeal(y^2,x^3);
o2 : MonomialIdeal of R
|
i3 : logCanonicalThreshold(I)
Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3
5
o3 = -
6
o3 : QQ
|
i4 : S = QQ[x,y,z];
|
i5 : J = monomialIdeal(x*y^4*z^6, x^5*y, y^7*z, x^8*z^8); -- Example 7 of [Howald 2000]
o5 : MonomialIdeal of S
|
i6 : logCanonicalThreshold(J)
Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3
68
o6 = ---
191
o6 : QQ
|
thresholds of multiplier ideals of monomial ideals
-
- Usage:
- logCanonicalThreshold(I,m)
-
Inputs:
-
Outputs:
-
a rational number, the least t such that m is not in the t-th multiplier ideal of I
-
a matrix, the equations of the facets of the Newton polyhedron of I which impose the threshold on m
Computes the threshold of inclusion of the monomial
m=xv in the multiplier ideal
J(It), that is, the value
t = sup{c | m lies in J(Ic) }= min{c | m does not lie in J(Ic)}. In other words,
(1/t)(v+(1,..,1)) lies on the boundary of the Newton polyhedron Newt(
I). In addition, returns the linear inequalities for those facets of Newt(
I) which contain
(1/t)(v+(1,..,1)). These are in the format of
Normaliz, i.e., a matrix
(A | b) where the number of columns of
A is the number of variables in the ring,
b is a column vector, and the inequality on the column vector
v is given by
Av+b ≥0, entrywise. As a special case, the log canonical threshold is the threshold of the monomial
1R = x0.
i7 : R = QQ[x,y];
|
i8 : I = monomialIdeal(x^13,x^6*y^4,y^9);
o8 : MonomialIdeal of R
|
i9 : logCanonicalThreshold(I,x^2*y)
Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3
1
o9 = (-, | 4 7 -52 |)
2 | 5 6 -54 |
o9 : Sequence
|
i10 : J = monomialIdeal(x^6,x^3*y^2,x*y^5); -- Example 6.7 of [Howald 2001] (thesis)
o10 : MonomialIdeal of R
|
i11 : logCanonicalThreshold(J,1_R)
Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3
5
o11 = (--, | 3 2 -13 |)
13
o11 : Sequence
|
i12 : logCanonicalThreshold(J,x^2)
Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3
3
o12 = (-, | 2 3 -12 |)
4
o12 : Sequence
|
log canonical threshold of a hyperplane arrangement
-
- Usage:
- logCanonicalThreshold A
-
Inputs:
-
Outputs:
Computes the log canonical threshold of a hyperplane arrangement
A.
i13 : R = QQ[x,y,z];
|
i14 : f = toList factor((x^2 - y^2)*(x^2 - z^2)*(y^2 - z^2)*z) / first;
|
i15 : A = arrangement f;
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i16 : logCanonicalThreshold(A)
3
o16 = -
7
o16 : QQ
|
log canonical threshold of monomial space curves
-
- Usage:
- logCanonicalThreshold(R,n)
-
Inputs:
-
R, a ring
-
n, a list, a list of three integers
-
Outputs:
Computes the log canonical threshold of the ideal I of a space curve parametrized by u →(ua,ub,uc).
i17 : R = QQ[x,y,z];
|
i18 : n = {2,3,4};
|
i19 : logCanonicalThreshold(R,n)
Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3
11
o19 = --
6
o19 : QQ
|
log canonical threshold of a generic determinantal ideal
-
- Usage:
- multiplierIdeal(L,r)
-
Inputs:
-
L, a list, dimensions {m,n} of a matrix
-
r, an integer, the size of minors generating the determinantal ideal
-
Outputs:
Computes the log canonical threshold of the ideal of
r ×r minors in a
m ×n matrix whose entries are independent variables (a generic matrix).
lct of ideal of 2-by-2 minors of 4-by-5 matrix:
i20 : x = getSymbol "x";
|
i21 : R = QQ[x_1..x_20];
|
i22 : X = genericMatrix(R,4,5);
4 5
o22 : Matrix R <--- R
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i23 : logCanonicalThreshold(X,2)
o23 = 10
o23 : QQ
|
We produce some tables of lcts:
i24 :
lctTable = (M,N,r) -> (
x = getSymbol "x";
R := QQ[x_1..x_(M*N)];
netList (
prepend( join({"m\\n"}, toList(3..M)),
for n from 3 to N list (
prepend(n,
for m from 3 to min(n,M) list (
logCanonicalThreshold(genericMatrix(R,m,n),r)
))
))
));
|
Table of LCTs of ideals of 3-by-3 minors of various size matrices (Table A.1 of [Johnson, 2003] (dissertation))
i25 :
lctTable(6,10,3)
+---+-+--+--+--+
o25 = |m\n|3|4 |5 |6 |
+---+-+--+--+--+
|3 |1| | | |
+---+-+--+--+--+
|4 |2|4 | | |
+---+-+--+--+--+
|5 |3|6 |8 | |
+---+-+--+--+--+
| | |15| | |
|6 |4|--|10|12|
| | | 2| | |
+---+-+--+--+--+
| | | |35| |
|7 |5|9 |--|14|
| | | | 3| |
+---+-+--+--+--+
| | |21|40| |
|8 |6|--|--|16|
| | | 2| 3| |
+---+-+--+--+--+
|9 |7|12|15|18|
+---+-+--+--+--+
| | |40|50| |
|10 |8|--|--|20|
| | | 3| 3| |
+---+-+--+--+--+
|
Table of LCTs of ideals of 4-by-4 minors of various size matrices (Table A.2 of [Johnson, 2003] (dissertation))
i26 : lctTable(8,14,4)
+---+-+--+--+--+--+--+
o26 = |m\n|3|4 |5 |6 |7 |8 |
+---+-+--+--+--+--+--+
|3 |0| | | | | |
+---+-+--+--+--+--+--+
|4 |0|1 | | | | |
+---+-+--+--+--+--+--+
|5 |0|2 |4 | | | |
+---+-+--+--+--+--+--+
|6 |0|3 |6 |8 | | |
+---+-+--+--+--+--+--+
| | | |15| | | |
|7 |0|4 |--|10|12| |
| | | | 2| | | |
+---+-+--+--+--+--+--+
| | | | |35| | |
|8 |0|5 |9 |--|14|16|
| | | | | 3| | |
+---+-+--+--+--+--+--+
| | | |21|40|63| |
|9 |0|6 |--|--|--|18|
| | | | 2| 3| 4| |
+---+-+--+--+--+--+--+
| | | | | |35| |
|10 |0|7 |12|15|--|20|
| | | | | | 2| |
+---+-+--+--+--+--+--+
| | | |40|33|77| |
|11 |0|8 |--|--|--|22|
| | | | 3| 2| 4| |
+---+-+--+--+--+--+--+
| | | |44| | | |
|12 |0|9 |--|18|21|24|
| | | | 3| | | |
+---+-+--+--+--+--+--+
| | | | |39|91| |
|13 |0|10|16|--|--|26|
| | | | | 2| 4| |
+---+-+--+--+--+--+--+
| | | |52| |49| |
|14 |0|11|--|21|--|28|
| | | | 3| | 2| |
+---+-+--+--+--+--+--+
|