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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 coefficientRing 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.
i1 : R=QQ[a,b,c,d,e,f]

o1 = R

o1 : PolynomialRing
i2 : M=matrix{{a,c,e},{b,d,f}}

o2 = | a c e |
     | b d f |

             2       3
o2 : Matrix R  <--- R
i3 : analyticSpread image M

o3 = 3
i4 : specialFiber image M

o4 = QQ[w , w , w ]
         0   1   2

o4 : PolynomialRing

See also

Ways to use specialFiber :