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KustinMiller :: unprojectionHomomorphism

unprojectionHomomorphism -- Compute the homomorphism associated to an unprojection pair

Synopsis

Description

Compute the deformation associated to the unprojection of I ⊂J (or equivalently of J’⊂R/I where R = ring I and J’=substitute(J,R/I)), i.e., a homomorphism

φ: J’ →R/I

such that the unprojected ideal U⊂R[T] is the inverse image of

U’ = (T*u - φ(u) | u ∈J’ ) ⊂(R/I)[T]

under the natural map R[T]→(R/I)[T].

The result is represented by a matrix f with source f = J’ and target f = (R/I)1.

i1 : R = QQ[x_1..x_4,z_1..z_4, T]

o1 = R

o1 : PolynomialRing
i2 : I = ideal(z_2*z_3-z_1*z_4,x_4*z_3-x_3*z_4,x_2*z_2-x_1*z_4,x_4*z_1-x_3*z_2,x_2*z_1-x_1*z_3)

o2 = ideal (z z  - z z , x z  - x z , x z  - x z , x z  - x z , x z  - x z )
             2 3    1 4   4 3    3 4   2 2    1 4   4 1    3 2   2 1    1 3

o2 : Ideal of R
i3 : J = ideal (z_1..z_4)

o3 = ideal (z , z , z , z )
             1   2   3   4

o3 : Ideal of R
i4 : phi = unprojectionHomomorphism(I,J)

o4 = | x_2x_4 x_2x_3 x_1x_4 x_1x_3 |

o4 : Matrix
i5 : S = ring target phi;
i6 : I == ideal S

o6 = true
i7 : source phi

o7 = image | z_4 z_3 z_2 z_1 |

                             1
o7 : S-module, submodule of S
i8 : target phi

      1
o8 = S

o8 : S-module, free

See also

Ways to use unprojectionHomomorphism :