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ReesAlgebra :: specialFiber

specialFiber -- Special fiber of a blowup

Synopsis

Description

Let M be an R = k[x1,...,xn]/J-module (for example an ideal), and let mm=ideal vars R = (x1,...,xn), and suppose that M is a homomorphic image of the free module F with m+1 generators. Let T be the Rees algebra of M. The call specialFiber(M) returns the ideal J⊂ k[w0,…,wm] such that k[w0,…,wm]/J ≅ T/mm*T; that is, specialFiber(M) = reesIdeal(M)+mm*Sym(F). This routine differs from specialFiberIdeal in that the ambient ring of the output ideal is k[w0,…,wm] rather than R[w0,…,wm]. The coefficient ring k used is always the ultimate coefficient ring of R.

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.

With the default Trim => true, the computation begins by computing minimal generators, which may result in a change of generators of M

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 : specialFiber minors(2,M)

     QQ[Z , Z , Z , Z , Z , Z ]
         0   1   2   3   4   5
o4 = --------------------------
         Z Z  - Z Z  + Z Z
          2 3    1 4    0 5

o4 : QuotientRing

See also

Ways to use specialFiber :