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MultiplierIdeals :: jumpingNumbers

jumpingNumbers -- jumping numbers

Synopsis

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.

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                                     2       
o3 = {{2, -, -, -, 3, --, -, --, 4}, {ideal (z, y), ideal (y, z*w, z , x*z),
          3  2  3      3  2   3
     ------------------------------------------------------------------------
                                 2                               2      
     ideal (z*w, y*w, y*z, x*z, y , x*y), ideal (y*w, y*z, x*z, y , x*y,
     ------------------------------------------------------------------------
        2   2                   2      2     2   2             2   2 2  
     z*w , z w), ideal (y*z*w, y w, y*z , x*z , y z, x*y*z, x*y , z w ),
     ------------------------------------------------------------------------
             2    2             2       2     2      2      3     3   2 2 
     ideal (y w, y z, x*y*z, x*y , y*z*w , y*z w, x*z w, y*z , x*z , x z ,
     ------------------------------------------------------------------------
      2 3   3 2               2   2 2     2      2    2               3  
     z w , z w ), ideal (y*z*w , y w , y*z w, x*z w, y z*w, x*y*z*w, y w,
     ------------------------------------------------------------------------
        2    2 2       2   2 2   3      2    2        3   2 2   2 3        
     x*y w, y z , x*y*z , x z , y z, x*y z, x y*z, x*y , x y , z w ), ideal
     ------------------------------------------------------------------------
       2 2   2               3      2    2 2       2   2 2   3      2  
     (y w , y z*w, x*y*z*w, y w, x*y w, y z , x*y*z , x z , y z, x*y z,
     ------------------------------------------------------------------------
      2        3   2 2       3     2 2     2 2     3      3    2 4   3 3  
     x y*z, x*y , x y , y*z*w , y*z w , x*z w , y*z w, x*z w, z w , z w ),
     ------------------------------------------------------------------------
             2   2   3 2   2 2        2    3        2        3    2 3 
     ideal (y z*w , y w , y z w, x*y*z w, y z*w, x*y z*w, x*y w, y z ,
     ------------------------------------------------------------------------
          3   2 3   3 2     2 2   2   2     3    2 2    2 3     2 3     3 2 
     x*y*z , x z , y z , x*y z , x y*z , x*y z, x y z, x y , y*z w , y*z w ,
     ------------------------------------------------------------------------
        3 2   3 4
     x*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.

References

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

See also

Ways to use jumpingNumbers :