parametrization -- compute a rational parametrization

Synopsis

Usage:

H=parametrization(PJ,adjList)
Inputs:
- PJ, a ring, coordinate ring of the last adjoint surface
- adjList, a list, list of adjoint matrices
Outputs:
- H, a matrix, parametrization of the rational surface X

Description

Let adjList be the list of adjoint matrices coming out of the adjunction process of a rational surface X. If the final surface is a P2 then the function computes the rational parametrization of X. In other cases the function returns rational parametrization from the final surface X'' in the adjunction process.

i1 : d=4

o1 = 4

i2 : L=toList(7:1)

o2 = {1, 1, 1, 1, 1, 1, 1}

o2 : List

i3 : n=expectedDimension(d,L)-1

o3 = 7

i4 : kk=ZZ/nextPrime(10^3)

o4 = kk

o4 : QuotientRing

i5 : t=symbol t, x= symbol x

o5 = (t, x)

o5 : Sequence

i6 : P2=kk[t_0..t_2]

o6 = P2

o6 : PolynomialRing

i7 : Pn=kk[x_0..x_n]

o7 = Pn

o7 : PolynomialRing

i8 : betti(I=rationalSurface(P2,d,L,Pn))

            0  1
o8 = total: 1 12
         0: 1  .
         1: . 12

o8 : BettiTally

i9 : minimalBetti I

            0  1  2  3  4 5
o9 = total: 1 12 25 21 10 3
         0: 1  .  .  .  . .
         1: . 12 25 15  . .
         2: .  .  .  6 10 3

o9 : BettiTally

i10 : (numList,adjList,ptsList,J)=adjunctionProcess(I);

i11 : numList

o11 = {(7, 9, 3), 7, (2, 1, 0)}

o11 : List

i12 : P2=ring J

o12 = P2

o12 : PolynomialRing

i13 : betti(H=parametrization(P2,adjList))

             0 1
o13 = total: 1 8
          0: 1 .
          1: . .
          2: . .
          3: . 8

o13 : BettiTally

i14 : elapsedTime sub(I,H)
 -- 0.0253419 seconds elapsed

o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

o14 : Ideal of P2

i15 : phi=map(P2,Pn,H);

o15 : RingMap P2 <-- Pn

i16 : elapsedTime betti(I'=trim ker phi)
 -- 0.119436 seconds elapsed

             0  1
o16 = total: 1 12
          0: 1  .
          1: . 12

o16 : BettiTally

i17 : I'== I

o17 = true

i18 : elapsedTime basePts=primaryDecomposition ideal H;
 -- 3.42742 seconds elapsed

i19 : tally apply(basePts,c->(dim c, degree c, betti c))

                           0  1
o19 = Tally{(0, 34, total: 1 15) => 1}
                        0: 1  .
                        1: .  .
                        2: .  .
                        3: .  8
                        4: .  .
                        5: .  7
                          0 1
            (1, 1, total: 1 2) => 7
                       0: 1 2

o19 : Tally

Ways to use parametrization :

parametrization(Ring,List)

For the programmer

The object parametrization is a method function.