specialFiberIdeal M
specialFiberIdeal(M,f)
Let $M$ be an $R = k[x_1,\ldots,x_n]/J$-module (for example an ideal), and let $mm=ideal vars R = (x_1,\ldots,x_n)$, and suppose that $M$ is a homomorphic image of the free module $F$. Let $T$ be the Rees algebra of $M$. The call specialFiberIdeal(M) returns the ideal $J\subset{} Sym(F)$ such that $Sym(F)/J \cong{} T/mm*T$; that is, $specialFiberIdeal(M) = reesIdeal(M)+mm*Sym(F).$
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
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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.
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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.
The object specialFiberIdeal is a method function with options.