getFlatFamily -- Compute the flat family depending on a subset of parameters of the universal unfolding

Synopsis

Usage:

(I,J1,family)=getFlatFamily(L,RR,ZZ)
Inputs:
- L, a list, the generators of a semigroup
- r, a real number,
- n, an integer, numbers which influences the truncation
Optional inputs:
- BaseField
- Verbose => ..., default value false
Outputs:
- I, an ideal, semigroup ideal
- J1, an ideal, flatness relations among the parameters
- family, a matrix, defining equation of the family

Description

After computing an unfolding and restricting the unfolding to variables of degree larger than

(maximal degree of a parameter)*r+n,

we compute the flattening relations and remove dependent variables. The remaining flattening relation are returned in the ideal J1. Using the function isARandomFiberSmooth we then can check with good luck whether a random fiber over some component of J1 is smooth.

i1 : L={6,8,10,11}

o1 = {6, 8, 10, 11}

o1 : List

i2 : genus L

o2 = 9

i3 : (I,J1,family)=getFlatFamily(L,0.30,0);

i4 : betti res ideal family == betti res I

o4 = true

i5 : isARandomFiberSmooth(I,J1,family)

o5 = true

i6 : support family

o6 = {x , x , x , x , a      , a      , a      , a      , a      , a      ,
       0   2   4   5   {3, 0}   {3, 1}   {3, 2}   {3, 3}   {1, 1}   {3, 4} 
     ------------------------------------------------------------------------
     a      , a      , a      , a      , a      , a      }
      {3, 5}   {1, 2}   {0, 1}   {3, 6}   {1, 3}   {0, 2}

o6 : List

i7 : support family /degree

o7 = {{6}, {8}, {10}, {11}, {22}, {16}, {14}, {12}, {12}, {11}, {10}, {10},
     ------------------------------------------------------------------------
     {10}, {8}, {8}, {8}}

o7 : List

i8 : gens ring J1 /degree

o8 = {{22}, {20}, {18}, {16}, {16}, {14}, {14}, {12}, {12}, {12}, {11}, {10},
     ------------------------------------------------------------------------
     {10}, {10}, {10}, {9}, {8}, {8}, {8}, {8}, {7}, {6}, {6}, {6}, {6}, {5},
     ------------------------------------------------------------------------
     {5}, {4}, {4}, {4}, {4}, {3}, {3}, {2}, {2}, {2}, {2}, {1}, {1}, {1}}

o8 : List

Parameters of the universal unfolding of degree <= 22*0.3 are not used

i9 : (I,J1,family)=getFlatFamily(L,0.00,11);

i10 : support family

o10 = {x , x , x , x , a      , a      , a      , a      , a      }
        0   2   4   5   {3, 0}   {3, 1}   {3, 2}   {3, 3}   {1, 1}

o10 : List

i11 : support family /degree

o11 = {{6}, {8}, {10}, {11}, {22}, {16}, {14}, {12}, {12}}

o11 : List

Parameters of the universal unfolding of degree < 11) are not used

i12 : isARandomFiberSmooth(I,J1,family)

o12 = true

i13 : A = ring family

o13 = A

o13 : PolynomialRing

i14 : transpose family

o14 = {-16} | x_2^2-x_0x_4                                                |
      {-18} | x_0^3-x_2x_4+x_0a_{1, 1}                                    |
      {-20} | x_0^2x_2-x_4^2+x_2a_{1, 1}                                  |
      {-22} | x_0^2x_4-x_5^2+a_{3, 0}+x_0a_{3, 1}+x_2a_{3, 2}+x_4a_{3, 3} |

              4      1
o14 : Matrix A  <-- A

Ways to use getFlatFamily:

getFlatFamily(List,RR,ZZ)

For the programmer

The object getFlatFamily is a method function with options.

getFlatFamily -- Compute the flat family depending on a subset of parameters of the universal unfolding

Synopsis

Description

See also

Ways to use getFlatFamily:

For the programmer