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ReesAlgebra :: specialFiberIdeal

specialFiberIdeal -- 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. Let T be the Rees algebra of M. The call specialFiberIdeal(M) returns the ideal J⊂ Sym(F) such that Sym(F)/J ≅ 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 , Z , Z , Z , Z ]
                  0   1   2   3   4   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 := dn, 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 , w ]
      32003  0   1   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.

See also

Ways to use specialFiberIdeal :