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getSmoothingFamily -- Get a smoothing family for the semigroup L

Description

We first test whether by the bound b or the range of degrees the restricted unfolding leads to a smoothing family over a finite field. If yes and with Verbose>0 then the degrees of the parameters of the parameters of the smoothing family are printed. In a second step we try to lift the example to characteristic 0 and check smoothness again.

i1 : L=(toDoList 8)_0

o1 = {6, 7, 9, 17}

o1 : List
i2 : (smooth,fib)=getSmoothingFamily(L,12,Verbose=>1)
    number of components = 1, codimension of components = {0}
    smoothing components numbers = {}

o2 = (false, )

o2 : Sequence
i3 : (smooth,fib)=getSmoothingFamily(L,11,Verbose=>1)
number of components = 2, codimension of components = {1, 2}
semigroup = {6, 7, 9, 17}
dim and degree singF = (0, 4)
smoothing components numbers = {1}

                    3    2      12   3    2        15   2               17
o3 = (true, ideal (x  - x  - x z  , x  - x x  - x z  , x x  - x x  - x z  ,
                0    3    0      1    0 3    0      1 3    0 5    0
    2             15      17   2 2           2 12      17     3    2
 x x  - x x  + x z   - x z  , x x  - x x  - x z   - x z  , x x  - x  +
  0 3    1 5    3       1      0 1    3 5    1       3      1 3    5
  2   15       17      27    34
 x x z   - 2x z   - x z   - z  ))
  0 1        5       1

o3 : Sequence
i4 : range=makeRange(L,{4})

o4 = {4, 8, 12, 16, 20, 24, 28, 32}

o4 : List
i5 : (smooth,fib, comps)=getSmoothingFamily(L,range,Verbose=>1)
time to decompose J1 :

component number = 0

component number = 1
deformation weights = {{12}, {8}, {8}, {4}, {4}, {4}}
semigroup = {6, 7, 9, 17}
dim and degree singF = (0, 8)
smoothing components numbers = {1}
flat = true

                    3    2    2 4      12   3    2        4        8      12
o5 = (true, ideal (x  - x  - x z  + x z  , x  - x x  - x z  - x x z  - x z  ,
                0    3    1      0      1    0 3    5      0 1      3
       2             2   4        8      16     2                8    2 12
      x x  - x x  - x x z  - x x z  - x z  , x x  - x x  - 2x x z  - x z  ,
       1 3    0 5    0 1      0 3      1      0 3    1 5     1 3      0
      2 2                  4     2 8      20     3    2          4         8
     x x  - x x  - 2x x x z  - 2x z  + x z  , x x  - x  - x x x z  - 3x x z
      0 1    3 5     0 1 3       3      0      1 3    5    0 1 5       3 5
              12     2 16      28
    - 3x x x z   - 3x z   + x z  ), {1})
        0 1 3        3       0

o5 : Sequence
i6 : (smooth,fib, comps)=getSmoothingFamily(L,range,Verbose=>2)
   time to decompose J1 :
   number of components = 2, codimension of components = {2, 1}

   component number = 0

   component number = 1
   deformation weights = {{12}, {8}, {8}, {4}, {4}, {4}}
   codim J3 = 0, numgens J3 = 0
   semigroup = {6, 7, 9, 17}
   dim and degree singF = (0, 8)
   dim and degree singF = (0, 8)
   smoothing components numbers = {1}
   flat = true

                    3    2    2 4      12   3    2        4        8      12
o6 = (true, ideal (x  - x  - x z  + x z  , x  - x x  - x z  - x x z  - x z  ,
                0    3    1      0      1    0 3    5      0 1      3
      2             2   4        8      16     2                8    2 12
     x x  - x x  - x x z  - x x z  - x z  , x x  - x x  - 2x x z  - x z  ,
      1 3    0 5    0 1      0 3      1      0 3    1 5     1 3      0
      2 2                  4     2 8      20     3    2          4         8
     x x  - x x  - 2x x x z  - 2x z  + x z  , x x  - x  - x x x z  - 3x x z
      0 1    3 5     0 1 3       3      0      1 3    5    0 1 5       3 5
              12     2 16      28
    - 3x x x z   - 3x z   + x z  ), {1})
        0 1 3        3       0

o6 : Sequence

The intermediate output is explained in smoothnessWithReductions.

See also

Ways to use getSmoothingFamily:

  • getSmoothingFamily(List,List)
  • getSmoothingFamily(List,ZZ)

For the programmer

The object getSmoothingFamily is a method function with options.


The source of this document is in /build/reproducible-path/macaulay2-1.26.06+ds/M2/Macaulay2/packages/WeierstrassSemigroups.m2:3175:0.