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

## Synopsis

• 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 .