specialFiber -- Special fiber of a blowup

Synopsis

Usage:

specialFiber M

specialFiber(M,f)
Inputs:
- M, a module, or an ideal
- f, a ring element, an optional element, which is a non-zerodivisor such that $M[f^{-1}]$ is a free module when $M$ is a module, an element in $M$ when $M$ is an ideal
Optional inputs:
- BasisElementLimit => ..., default value infinity, Bound the number of Groebner basis elements to compute in the saturation step
- DegreeLimit => ..., default value {}, Bound the degrees considered in the saturation step. Defaults to infinity
- Jacobian => ..., default value false
- MinimalGenerators => ..., default value true, Whether the saturation step returns minimal generators
- PairLimit => ..., default value infinity, Bound the number of s-pairs considered in the saturation step
- Strategy => ..., default value null, Choose a strategy for the saturation step
- Trim => ..., default value true
- Variable => ..., default value "w", Choose name for variables in the created ring
Outputs:
- a ring,

Description

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 $F$ with $m+1$ generators. Let $T$ be the Rees algebra of $M$. The call specialFiber(M) returns the ideal $J\subset{} k[w_0,\dots,w_m]$ such that $k[w_0,\dots,w_m]/J \cong{} T/mm*T$; that is, $specialFiber(M) = reesIdeal(M)+mm*Sym(F)$. This routine differs from specialFiberIdeal in that the ambient ring of the output ideal is $k[w_0,\dots,w_m]$ rather than $R[w_0,\dots,w_m]$. 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 ]
             0   5
o4 = ------------------
     Z Z  - Z Z  + Z Z
      2 3    1 4    0 5

o4 : QuotientRing

Ways to use specialFiber :

specialFiber(Ideal)
specialFiber(Ideal,RingElement)
specialFiber(Module)
specialFiber(Module,RingElement)

For the programmer

The object specialFiber is a method function with options.

specialFiber -- Special fiber of a blowup

Synopsis

Description

See also

Ways to use specialFiber :

For the programmer