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.
R = QQ[x,y]; |
I = monomialIdeal(y^2,x^3); |
logCanonicalThreshold(I) |
S = QQ[x,y,z]; |
J = monomialIdeal(x*y^4*z^6, x^5*y, y^7*z, x^8*z^8); -- Example 7 of [Howald 2000] |
logCanonicalThreshold(J) |
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.
R = QQ[x,y]; |
I = monomialIdeal(x^13,x^6*y^4,y^9); |
logCanonicalThreshold(I,x^2*y) |
J = monomialIdeal(x^6,x^3*y^2,x*y^5); -- Example 6.7 of [Howald 2001] (thesis) |
logCanonicalThreshold(J,1_R) |
logCanonicalThreshold(J,x^2) |
log canonical threshold of a hyperplane arrangement
- Usage:
- logCanonicalThreshold A
- Inputs:
- Outputs:
Computes the log canonical threshold of a hyperplane arrangement
A.
R = QQ[x,y,z]; |
f = toList factor((x^2 - y^2)*(x^2 - z^2)*(y^2 - z^2)*z) / first; |
A = arrangement f; |
logCanonicalThreshold(A) |
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).
R = QQ[x,y,z]; |
n = {2,3,4}; |
logCanonicalThreshold(R,n) |
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:
x = getSymbol "x"; |
R = QQ[x_1..x_20]; |
X = genericMatrix(R,4,5); |
logCanonicalThreshold(X,2) |
We produce some tables of lcts:
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))
Table of LCTs of ideals of 4-by-4 minors of various size matrices (Table A.2 of [Johnson, 2003] (dissertation))