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

## Synopsis

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