satSpecialFiberIdeal -- computes the defining equations of the saturated special fiber ring

Usage:

satSpecialFiberIdeal(I, nsteps)

satSpecialFiberIdeal(I)
Inputs:
- I, an ideal, a homogeneous ideal generated by elements of the same degree
- nteps, an integer, the number steps in the saturation of the powers of I. Optional.
Outputs:
- an ideal, the defining equations of the saturated special fiber ring

Description

The purpose of this function is to compute the defining equations of the special fiber ring.

Suppose that $\{g_1,...,g_m\}$ is the set of generators of the saturated special fiber ring (which can be obtained from "gensSatSpecialFib"). This function returns the kernel of the map $k[z_1, ... ,z_m] \to k[g_1, ... ,g_m]$ which is given by $$ z_i \to g_i. $$

First, we compute some examples of plane rational maps.

i1 : R = QQ[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : A = matrix{ {x, x^5 + y^5},
                 {-y, y^5 + z*x^2*y^2},
                 {0, x^5}
               };

             3      2
o2 : Matrix R  <-- R

i3 : I = minors(2, A) -- a birational map

             5       5    6    3 2    6    5
o3 = ideal (x y + x*y  + y  + x y z, x , -x y)

o3 : Ideal of R

i4 : satSpecialFiberIdeal I

o4 = ideal ()

o4 : Ideal of QQ[Z ..Z ]
                  1   3

i5 : A = matrix{ {x^3, x^2 + y^2},
                 {-y^3, y^2 + z*x},
                 {0, x^2}
               };

             3      2
o5 : Matrix R  <-- R

i6 : I = minors(2, A) -- a non birational map

             3 2    2 3    5    4    5    2 3
o6 = ideal (x y  + x y  + y  + x z, x , -x y )

o6 : Ideal of R

i7 : satSpecialFiberIdeal I

             2          2            2                   4          2 3  
o7 = ideal (Z  - Z Z , Z Z Z  - Z Z Z  - Z Z  + Z Z , Z Z  - Z Z , Z Z  +
             4    3 6   1 2 3    1 2 3    3 5    1 6   1 3    4 6   1 3  
     ------------------------------------------------------------------------
      2                           3      2   2 2      2                 
     Z Z Z  - Z Z Z Z  - Z Z , Z Z Z  - Z , Z Z Z  + Z Z Z  - Z Z Z Z  -
      1 2 4    1 2 3 4    4 5   1 3 4    6   1 3 4    1 2 6    1 2 3 6  
     ------------------------------------------------------------------------
            2 2        2        3                 2              4 2      2 3
     Z Z , Z Z Z  - Z Z Z Z  + Z Z  - Z Z Z  - Z Z Z  + Z Z Z , Z Z  - Z Z Z 
      5 6   1 2 4    1 2 3 4    3 5    2 4 5    1 3 6    2 4 6   1 2    1 2 3
     ------------------------------------------------------------------------
        3         2          2      2     2                  2        2 2    
     - Z Z Z  - 2Z Z Z  - Z Z Z  + Z  + 2Z Z Z  + Z Z Z Z , Z Z Z  + Z Z Z  -
        1 3 4     1 2 5    2 3 5    5     1 2 6    1 2 3 6   3 4 5    1 2 6  
     ------------------------------------------------------------------------
        2                              2   4        3
     Z Z Z Z  - Z Z Z Z  - Z Z Z  + Z Z , Z Z  - Z Z Z  - Z Z Z Z  +
      1 2 3 6    1 3 4 6    2 5 6    2 6   3 5    1 3 6    1 2 4 6  
     ------------------------------------------------------------------------
     Z Z Z Z )
      2 3 4 6

o7 : Ideal of QQ[Z ..Z ]
                  1   6

Next, we test some bigraded rational maps.

i8 : R = QQ[x,y,u,v, Degrees => {{1,0}, {1,0}, {0,1}, {0,1}}]

o8 = R

o8 : PolynomialRing

i9 : I = ideal(x*u, y*u, y*v) -- a birational map

o9 = ideal (x*u, y*u, y*v)

o9 : Ideal of R

i10 : satSpecialFiberIdeal(I, 5)

o10 = ideal ()

o10 : Ideal of QQ[Z ..Z ]
                   1   3

i11 : I = ideal(x*u, y*v, x*v + y*u) -- a non birational map

o11 = ideal (x*u, y*v, y*u + x*v)

o11 : Ideal of R

i12 : satSpecialFiberIdeal(I, 5)

o12 = ideal(Z Z  - Z Z )
             2 3    1 4

o12 : Ideal of QQ[Z ..Z ]
                   1   4

Caveat

To call the method "satSpecialFiberIdeal(I)", the ideal $I$ should be in a single graded polynomial ring.

The answer of "satSpecialFiberIdeal(I, nsteps)" is correct only if nsteps is big enough to attain all the generators of the saturated special fiber ring.

Ways to use satSpecialFiberIdeal:

satSpecialFiberIdeal(Ideal,ZZ)
satSpecialFiberIdeal(Ideal) (missing documentation)

For the programmer

The object satSpecialFiberIdeal is a method function.

The source of this document is in MultiGradedRationalMap.m2:962:0.