gensSatSpecialFib -- computes generators of the saturated special fiber ring

Synopsis

Usage:

gensSatSpecialFib(I, nsteps)

gensSatSpecialFib(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:
- a list, a list of generators of the saturated special fiber ring. In the case where we use the function as "gensSatSpecialFib(I, nsteps)", the answer is correct only if nsteps is big enough to attain all the generators.

Description

This function computes generators of the saturated special fiber ring.

When we call "gensSatSpecialFib(I, nsteps)", the method iteratively computes the graded pieces $$ [(I^1)^{sat}]_d, [(I^2)^{sat}]_{2d}, ......... , [(I^{nsteps})^{sat}]_{nsteps*d}, $$ where $(I^k)^{sat}$ denotes the saturation of $I$ with respect to the irrelevant ideal.

When we call "gensSatSpecialFib(I)", the method first computes the module $[H_m^1(Rees(I))]_0$ from which an upper bound nsteps. After that, it simply calls "gensSatSpecialFib(I, nsteps)".

First, we compute some examples in the case of plane rational maps.

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

o1 = R

o1 : PolynomialRing

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

             3      2
o2 : Matrix R  <-- R

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

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

o3 : Ideal of R

i4 : gensSatSpecialFib I

         2    3    2    2    3
o4 = {x*y  + y  + x z, x y, x }

o4 : List

i5 : gensSatSpecialFib(I, 5)

         2    3    2    2    3
o5 = {x*y  + y  + x z, x y, x }

o5 : List

i6 : A = matrix{ {x^3, x^2 + y^2},
                 {-y^3, y^2 + z*x},
                 {0, x^2}
               };

             3      2
o6 : Matrix R  <-- R

i7 : I = minors(2, A) -- a non birational map

             3 2    2 3    5    4    5    2 3
o7 = ideal (x y  + x y  + y  + x z, x , -x y )

o7 : Ideal of R

i8 : gensSatSpecialFib I

       2 3   3 2    5    4    5   9    4 11    11 3     8 6    13 2
o8 = {x y , x y  + y  + x z, x , x y, x y   - x  y z + x y z, x  y }

o8 : List

i9 : gensSatSpecialFib(I, 5)

       2 3   3 2    5    4    5   9    4 11    11 3     8 6    13 2
o9 = {x y , x y  + y  + x z, x , x y, x y   - x  y z + x y z, x  y }

o9 : List

Next, we compute an example in the bigraded case.

i10 : R = QQ[x,y,u,v, Degrees => {{1,0}, {1,0}, {0,1}, {0,1}}]

o10 = R

o10 : PolynomialRing

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 : gensSatSpecialFib(I, 5)

o12 = {x*u, x*v, y*u, y*v}

o12 : List

Caveat

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

Ways to use gensSatSpecialFib :

gensSatSpecialFib(Ideal)
gensSatSpecialFib(Ideal,ZZ)

For the programmer

The object gensSatSpecialFib is a method function.