The purpose of this function is to compute the defining equations of the special fiber ring.
Suppose that {g1,...,gm} 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[z1, ... ,zm] →k[g1, ... ,gm] which is given by
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 , Z ]
1 2 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 2 3 2 2 2 2
o7 = ideal (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 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 1 2 4 1 2 3 4 4 5 1 3 4 6 1 3 4 1 2 6
----------------------------------------------------------------------------------------------------------------------------
2 2 2 3 2 4 2 2 3 3 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 Z , Z Z - Z Z Z - Z Z Z - 2Z Z Z - Z Z Z + Z +
1 2 3 6 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 1 3 4 1 2 5 2 3 5 5
----------------------------------------------------------------------------------------------------------------------------
2 2 2 2 2 2 4 3
2Z 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 , Z Z - Z Z Z - Z Z Z Z + Z Z Z Z )
1 2 6 1 2 3 6 3 4 5 1 2 6 1 2 3 6 1 3 4 6 2 5 6 2 6 3 5 1 3 6 1 2 4 6 2 3 4 6
o7 : Ideal of QQ[Z , Z , Z , Z , Z , Z ]
1 2 3 4 5 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 , Z ]
1 2 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 , Z , Z ]
1 2 3 4
|
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.