- Usage:
`specialFiberIdeal M``specialFiberIdeal(M,f)`

- Inputs:
`M`, a module, or an ideal`f`, a ring element, a non-zerodivisor such that*M[f*is a free module when^{-1}]*M*is a module, an element in*M*when*M*is an ideal

- Optional inputs:
- BasisElementLimit => ...,
- DegreeLimit => ...,
`Jacobian => ...`(missing documentation),- MinimalGenerators => ...,
- PairLimit => ...,
- Strategy => ...,
- Variable => ...,

- Outputs:
- an ideal

Let *M* be an *R = k[x _{1},...,x_{n}]/J*-module (for example an ideal), and let

The name derives from the fact that *Proj(T/mm*T)* is the special fiber of the blowup of *Spec R* along the subscheme defined by *I*.

i1 : R=QQ[a..h] o1 = R o1 : PolynomialRing |

i2 : M=matrix{{a,b,c,d},{e,f,g,h}} o2 = | a b c d | | e f g h | 2 4 o2 : Matrix R <--- R |

i3 : analyticSpread minors(2,M) o3 = 5 |

i4 : specialFiberIdeal minors(2,M) o4 = ideal(Z Z - Z Z + Z Z ) 2 3 1 4 0 5 o4 : Ideal of QQ[Z , Z , Z , Z , Z , Z ] 0 1 2 3 4 5 |

If M is an n x n+1 matrix in n variables, and all generators have the same degree d, with ell = n as expected, then the special fiber is a rational hypersurface of degree *D := d ^{n}*, and the reduction number is D-1.

i5 : n = 2 o5 = 2 |

i6 : x = symbol x o6 = x o6 : Symbol |

i7 : S = ZZ/32003[x_1..x_n] o7 = S o7 : PolynomialRing |

i8 : M = matrix{{x_1,x_2,0},{0,x_1,x_2}} o8 = | x_1 x_2 0 | | 0 x_1 x_2 | 2 3 o8 : Matrix S <--- S |

i9 : I = minors(n,M) 2 2 o9 = ideal (x , x x , x ) 1 1 2 2 o9 : Ideal of S |

i10 : specialFiber(I,I_0) ZZ -----[w , w , w ] 32003 0 1 2 o10 = ----------------- 2 w - w w 1 0 2 o10 : QuotientRing |

Special fiber is here defined to be the fiber of the blowup over the subvariety defined by the vars of the original ring. Note that if the original ring is a tower ring, this might not be the fiber over the closed point! To get the closed fiber, flatten the base ring first.

- reesIdeal -- Compute the defining ideal of the Rees Algebra

- specialFiberIdeal(Ideal)
- specialFiberIdeal(Ideal,RingElement)
- specialFiberIdeal(Module)
- specialFiberIdeal(Module,RingElement)