specialFiberIdeal -- Special fiber of a blowup

Synopsis

Usage:

specialFiberIdeal M

specialFiberIdeal(M,f)
Inputs:
- M, a module, or an ideal
- f, a ring element, 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:
- an ideal

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 $F$. 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

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 ]
                  0   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 ]
      32003  0   2
o10 = -------------
         2
        w  - w w
         1    0 2

o10 : QuotientRing

Caveat

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.

Ways to use `specialFiberIdeal` :

"specialFiberIdeal(Ideal)"
"specialFiberIdeal(Ideal,RingElement)"
"specialFiberIdeal(Module)"
"specialFiberIdeal(Module,RingElement)"

For the programmer

The object specialFiberIdeal is a method function with options.