# logCanonicalThreshold -- log canonical threshold

## Description

The log canonical threshold of an ideal $I$ is the infimum of $t$ for which the multiplier ideal $J(I^t)$ is a proper ideal. Equivalently it is the least nonzero jumping number.

## log canonical threshold of a monomial ideal

• Usage:
logCanonicalThreshold I
• Inputs:
• I, ,
• 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) 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) 68 o6 = --- 191 o6 : QQ

## thresholds of multiplier ideals of monomial ideals

• Usage:
logCanonicalThreshold(I,m)
• Inputs:
• I, ,
• m, , a monomial
• Outputs:
• , the least $t$ such that $m$ is not in the $t$-th multiplier ideal of $I$
• , 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 $m=x^v$ in the multiplier ideal $J(I^t)$, that is, the value $t = sup\{ c | m lies in J(I^c) \} = min\{ c | m does not lie in J(I^c)\}$. 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 $Av+b \geq 0$, entrywise. As a special case, the log canonical threshold is the threshold of the monomial $1_R = x^0$.
 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) 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) 5 o11 = (--, | 3 2 -13 |) 13 o11 : Sequence i12 : logCanonicalThreshold(J,x^2) 3 o12 = (-, | 2 3 -12 |) 4 o12 : Sequence

## log canonical threshold of a hyperplane arrangement

• Usage:
logCanonicalThreshold A
• Inputs:
• A, , a central hyperplane arrangement
• 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; 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:
• logCanonicalThreshold,

Computes the log canonical threshold of the ideal $I$ of a space curve parametrized by $u \to (u^a,u^b,u^c)$.

 i17 : R = QQ[x,y,z]; i18 : n = {2,3,4}; i19 : logCanonicalThreshold(R,n) 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 $r \times r$ minors in a $m \times n$ 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 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| | +---+-+--+--+--+--+--+