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