rationalSurface -- compute the ideal I of the rational surface

Synopsis

Usage:

I=rationalSurface(P2,d,L,Pn)

I=rationalSurface(P2,d,L,x)

I=rationalSurface(points,d,L,x)
Inputs:
- P2, a ring, homogeneous coordinate ring of P2,
- points, a list, list of ideals of points in P2
- d, an integer, degree of the desired forms
- L, a list, {r_1,...,r_s} of multiplicities
- Pn, a ring, coordinate ring of Pn
- x, a symbol, variable name
Outputs:
- I, an ideal, of a rational surface

Description

The function chooses randomly s point p_i in P2 and computes the linear system H=L(d;L) of form of degree d which have multiplicity r_i at the point p_i.

Then the ideal of the image of P2 of under the rational map defined by the linear system H is computed.

i1 : d=6

o1 = 6

i2 : L=toList(6:2)|toList(2:1)

o2 = {2, 2, 2, 2, 2, 2, 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

o5 = t

o5 : Symbol

i6 : P2=kk[t_0..t_2]

o6 = P2

o6 : PolynomialRing

i7 : y=symbol y

o7 = y

o7 : Symbol

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

            0  1
o8 = total: 1 11
         0: 1  .
         1: . 11

o8 : BettiTally

i9 : x = symbol x

o9 = x

o9 : Symbol

i10 : Pn=kk[x_0..x_n]

o10 = Pn

o10 : PolynomialRing

i11 : betti(I=rationalSurface(P2,d,L,Pn))

             0  1
o11 = total: 1 11
          0: 1  .
          1: . 11

o11 : BettiTally

i12 : degree I, genus I, dim I

o12 = (10, 0, 3)

o12 : Sequence

i13 : minimalBetti I

             0  1  2  3  4 5
o13 = total: 1 11 24 25 15 4
          0: 1  .  .  .  . .
          1: . 11 20  9  . .
          2: .  .  4 16 15 4

o13 : BettiTally

i14 : d^2-sum(L,r->r^2)== degree I

o14 = true

i15 : (numList,adjList,ptsList,J)=adjunctionProcess(I);

i16 : numList

o16 = {(7, 10, 4), 2, (3, 3, 1)}

o16 : List

i17 : minimalBetti J

             0 1
o17 = total: 1 1
          0: 1 .
          1: . .
          2: . 1

o17 : BettiTally

Ways to use rationalSurface :

rationalSurface(List,ZZ,List,Symbol)
rationalSurface(Ring,ZZ,List,Ring)
rationalSurface(Ring,ZZ,List,Symbol)

For the programmer

The object rationalSurface is a method function.