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
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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
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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
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i4 : satSpecialFiberIdeal I
o4 = ideal ()
o4 : Ideal of QQ[Z ..Z ]
1 3
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i5 : A = matrix{ {x^3, x^2 + y^2},
{-y^3, y^2 + z*x},
{0, x^2}
};
3 2
o5 : Matrix R <-- R
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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
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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
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i8 : R = QQ[x,y,u,v, Degrees => {{1,0}, {1,0}, {0,1}, {0,1}}]
o8 = R
o8 : PolynomialRing
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i9 : I = ideal(x*u, y*u, y*v) -- a birational map
o9 = ideal (x*u, y*u, y*v)
o9 : Ideal of R
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i10 : satSpecialFiberIdeal(I, 5)
o10 = ideal ()
o10 : Ideal of QQ[Z ..Z ]
1 3
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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
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i12 : satSpecialFiberIdeal(I, 5)
o12 = ideal(Z Z - Z Z )
2 3 1 4
o12 : Ideal of QQ[Z ..Z ]
1 4
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