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) Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 warning: clearing value of symbol w to allow access to subscripted variables based on it : debug with expression debug 365 or with command line option --debug 365 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 7 5 8 10 7 11 2 2 o3 = {{2, -, -, -, 3, --, -, --, 4}, {monomialIdeal (y, z), monomialIdeal (y, x*z, z , z*w), monomialIdeal (x*y, y , x*z, y*z, 3 2 3 3 2 3 ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 2 2 2 2 y*w, z*w), monomialIdeal (x*y, y , x*z, y*z, y*w, z w, z*w ), monomialIdeal (x*y , x*y*z, y z, x*z , y*z , y w, y*z*w, ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 2 2 3 3 2 2 2 2 3 2 2 3 2 2 3 z w ), monomialIdeal (x*y , x*y*z, y z, x z , x*z , y*z , y w, x*z w, y*z w, y*z*w , z w , z w ), monomialIdeal (x y , x*y , ---------------------------------------------------------------------------------------------------------------------------- 2 2 3 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 x y*z, x*y z, y z, x z , x*y*z , y z , x*y w, y w, x*y*z*w, y z*w, x*z w, y*z w, y w , y*z*w , z w ), monomialIdeal (x y , ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 3 2 2 2 2 2 2 3 2 3 3 2 2 2 2 2 2 3 3 3 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, y z*w, x*z w, y*z w, y w , x*z w , y*z w , y*z*w , z w , ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 3 2 2 3 2 2 2 2 3 2 2 3 3 2 3 3 2 3 2 2 2 z w ), monomialIdeal (x y , x y z, x*y z, x y*z , x*y z , y z , x z , x*y*z , y z , x*y w, x*y z*w, y z*w, x*y*z w, y z w, ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 3 2 3 2 2 3 3 4 y w , y z*w , x*z w , 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") Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 warning: clearing value of symbol w to allow access to subscripted variables based on it : debug with expression debug 365 or with command line option --debug 365 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 Using Normaliz integer input types is deprecated, please use rees_algebra instead of 3 Using Normaliz integer input types is deprecated, please use inequalities instead of 4 7 5 8 2 2 2 o6 = {{-, -, -, 3}, {monomialIdeal (y, x*z, z , z*w), monomialIdeal (x*y, y , x*z, y*z, y*w, z*w), monomialIdeal (x*y, y , x*z, 3 2 3 ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 y*z, y*w, z w, z*w ), monomialIdeal (x*y , x*y*z, y z, x*z , y*z , y w, y*z*w, z w )}} o6 : List |
The object jumpingNumbers is a function closure.