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# slowAdjunctionCalculation -- compute the adjoint variety and the presentation of O(1)

- Usage:
    `(I1,adj)=slowAdjunctionCalculation(I,D,z)`
- Inputs:
  - I, an ideal, of a smooth projective surface X
  - D, a matrix, the transpose of a linear presentation matrix of omega_X(1)
  - z, a symbol, variable name
- Outputs:
  - I1, an ideal, of the adjoint surface in a Pn
  - adj, a matrix, the presentation matrix of O_X(1) as O_Pn-module

## Description

compute the ideal of the adjoint variety and the presentation matrix of the line bundle O_X(1) as a module on the adjoint variety

    i1 : kk=ZZ/101

    o1 = kk

    o1 : QuotientRing

    i2 : P2=kk[x_0..x_2]

    o2 = P2

    o2 : PolynomialRing

    i3 : betti(Y=rationalSurface(P2,8,toList(4:3)|toList(4:2)|{1,1},symbol z))

                0 1
    o3 = total: 1 7
             0: 1 .
             1: . 5
             2: . 2

    o3 : BettiTally

    i4 : P6=ring Y, dim P6==7

    o4 = (P6, true)

    o4 : Sequence

    i5 : betti(fY=res Y)

                0 1  2  3 4
    o5 = total: 1 7 17 16 5
             0: 1 .  .  . .
             1: . 5  2  . .
             2: . 2 15 16 5

    o5 : BettiTally

    i6 : betti(omegaY=coker transpose fY.dd_4**P6^{-dim P6})

                0  1
    o6 = total: 5 16
             1: 5 16

    o6 : BettiTally

    i7 : betti(D=transpose presentation omegaY)

                 0 1
    o7 = total: 16 5
            -2: 16 5

    o7 : BettiTally

    i8 : (I,adj)=slowAdjunctionCalculation(Y,D,symbol x);

    i9 : betti adj

                0  1
    o9 = total: 7 16
             0: 7 16

    o9 : BettiTally

    i10 : P4=ring I, dim P4==5

    o10 = (P4, true)

    o10 : Sequence

    i11 : PI=P4/I

    o11 = PI

    o11 : QuotientRing

    i12 : m=adj**PI;

                   7       16
    o12 : Matrix PI  <-- PI

    i13 : betti(sm=syz transpose m)

                 0 1
    o13 = total: 7 2
              0: 7 .
              1: . 2

    o13 : BettiTally

    i14 : rank sm==1

    o14 = true

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

    i16 : numList, minimalBetti I

                                              0 1 2
    o16 = ({(6, 10, 5), 2, (4, 5, 2)}, total: 1 3 2)
                                           0: 1 . .
                                           1: . 1 .
                                           2: . 2 2

    o16 : Sequence

## Ways to use slowAdjunctionCalculation:

- `slowAdjunctionCalculation(Ideal,Matrix,Symbol)`

## For the programmer

The object slowAdjunctionCalculation is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/AdjunctionForSurfaces.m2:598:0.