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# rationalSurface -- compute the ideal I of the rational surface

- 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/AdjunctionForSurfaces.m2:513:0.