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# parametrization -- compute a rational parametrization

## Synopsis

• Usage:
H=parametrization(PJ,adjList)
• Inputs:
• PJ, a ring, coordinate ring of the last adjoint surface
• Outputs:
• H, , 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 .