jumpingNumbers -- jumping numbers

Description

Jumping numbers of an ideal $I$ are those real numbers $t$ at which the multiplier ideal $J(I^t)$, as a function of the parameter $t$, changes. More precisely, $t$ is a jumping number if $J(I^t)$ is different from $J(I^{t-\epsilon})$ for all $\epsilon > 0$. The jumping numbers form a discrete sequence of rational numbers. Thus $t_1, t_2$ are two consecutive jumping numbers of $I$ when $J(I^t) = J(I^{t_1})$ for all $t_1 \leq t < t_2$ and $J(I^t) \neq J(I^{t_1})$ for $t < t_1$ or $t_2 \leq t$.

The jumpingNumbers command determines the jumping numbers of an ideal along with the multiplier ideals at those jumping numbers. By definition, the multiplier ideals are then determined at the intermediate parameter values.

The jumpingNumbers command can handle any input that the multiplierIdeals command can handle (but omit the rational number argument of multiplierIdeals). That is, jumpingNumbers can handle the following inputs: a hyperplane arrangement; a hyperplane arrangement with a list of multiplicities; a monomial ideal; a monomial space curve (specified by giving a ring together with a list of exponents); a generic determinantal ideal (specified by giving a generic matrix together with a size of minors).

 i1 : R = QQ[x,y,z,w]; i2 : I = monomialIdeal(x*y, x*z, y*z, y*w, z*w^2); o2 : MonomialIdeal of R i3 : jumpingNumbers(I) 7 5 8 10 7 11 o3 = {{2, -, -, -, 3, --, -, --, 4}, {monomialIdeal (y, z), monomialIdeal (y, 3 2 3 3 2 3 ------------------------------------------------------------------------ 2 2 x*z, z , z*w), monomialIdeal (x*y, y , x*z, y*z, y*w, z*w), ------------------------------------------------------------------------ 2 2 2 2 monomialIdeal (x*y, y , x*z, y*z, y*w, z w, z*w ), monomialIdeal (x*y , ------------------------------------------------------------------------ 2 2 2 2 2 2 2 x*y*z, y z, x*z , y*z , y w, y*z*w, z w ), monomialIdeal (x*y , x*y*z, ------------------------------------------------------------------------ 2 2 2 3 3 2 2 2 2 3 2 2 3 y z, x z , x*z , y*z , y w, x*z w, y*z w, y*z*w , z w , z w ), ------------------------------------------------------------------------ 2 2 3 2 2 3 2 2 2 2 2 2 monomialIdeal (x y , x*y , x y*z, x*y z, y z, x z , x*y*z , y z , x*y w, ------------------------------------------------------------------------ 3 2 2 2 2 2 2 2 3 y w, x*y*z*w, y z*w, x*z w, y*z w, y w , y*z*w , z w ), monomialIdeal ------------------------------------------------------------------------ 2 2 3 2 2 3 2 2 2 2 2 2 3 (x y , x*y , x y*z, x*y z, y z, x z , x*y*z , y z , x*y w, y w, x*y*z*w, ------------------------------------------------------------------------ 2 3 3 2 2 2 2 2 2 3 3 3 2 4 y z*w, x*z w, y*z w, y w , x*z w , y*z w , y*z*w , z w , z w ), ------------------------------------------------------------------------ 2 3 2 2 3 2 2 2 2 3 2 2 3 3 monomialIdeal (x y , x y z, x*y z, x y*z , x*y z , y z , x z , x*y*z , ------------------------------------------------------------------------ 2 3 3 2 3 2 2 2 3 2 2 2 3 2 y z , x*y w, x*y z*w, y z*w, x*y*z w, y z w, y w , y z*w , x*z w , ------------------------------------------------------------------------ 3 2 2 3 3 4 y*z w , y*z w , z w )}} o3 : List

By default, jumpingNumbers looks for jumping numbers in a closed interval $[a,b]$ where $a$ is the log canonical threshold of the ideal and $b$ is a sufficiently large value to ensure that Skoda periodicity holds, that is, $J(I^t) = I J(I^{t-1})$ for $t \geq b$. (In particular, the multiplier ideals and jumping numbers are determined for all $t$ by the output of this command.) The user may specify a different interval via the optional arguments Interval and IntervalType.

 i4 : R = QQ[x,y,z,w]; i5 : I = monomialIdeal(x*y, x*z, y*z, y*w, z*w^2); o5 : MonomialIdeal of R i6 : jumpingNumbers(I,Interval=>{2,3},IntervalType=>"OpenClosed") 7 5 8 2 2 o6 = {{-, -, -, 3}, {monomialIdeal (y, x*z, z , z*w), monomialIdeal (x*y, y , 3 2 3 ------------------------------------------------------------------------ 2 2 2 x*z, y*z, y*w, z*w), monomialIdeal (x*y, y , x*z, y*z, y*w, z w, z*w ), ------------------------------------------------------------------------ 2 2 2 2 2 2 2 monomialIdeal (x*y , x*y*z, y z, x*z , y*z , y w, y*z*w, z w )}} o6 : List

References

• [ELSV] Ein, Lawrence, Robert Lazarsfeld, Karen E. Smith, and Dror Varolin, Jumping coefficients of multiplier ideals. Duke Math. J. 123 (2004), no. 3, 469-506.