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## Synopsis

• Usage:
adj=adjointMatrix(D,z)
• Inputs:
• D, , the transpose of a linear presentation matrix of omega_X(1)
• Outputs:
• adj, , the presentation matrix of O_X(1)

## Description

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

 i1 : d=7 o1 = 7 i2 : L=toList(7:2)|toList(8:1) o2 = {2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1} o2 : List i3 : n=expectedDimension(d,L)-1 o3 = 6 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 14 0: 1 . 1: . 2 2: . 12 o8 : BettiTally i9 : c=codim I o9 = 4 i10 : elapsedTime fI=res I -- 0.0380266 seconds elapsed 1 14 33 28 8 o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn <-- 0 0 1 2 3 4 5 o10 : ChainComplex i11 : betti(omega=presentation coker transpose fI.dd_c**Pn^{-n-1}) 0 1 o11 = total: 8 28 1: 8 28 o11 : BettiTally i12 : D=transpose omega; 28 8 o12 : Matrix Pn <-- Pn i13 : z= symbol z o13 = z o13 : Symbol i14 : betti(adj=adjointMatrix(D,z)) 0 1 o14 = total: 7 28 0: 7 28 o14 : BettiTally i15 : support adj o15 = {z , z , z , z , z , z , z , z } 0 1 2 3 4 5 6 7 o15 : List i16 : minimalBetti ann coker adj 0 1 2 3 4 5 o16 = total: 1 12 25 21 10 3 0: 1 . . . . . 1: . 12 25 15 . . 2: . . . 6 10 3 o16 : BettiTally i17 : (numList,adjList,ptsList,J)=adjunctionProcess(I,1); i18 : numList o18 = {(6, 13, 8), 8, (7, 9, 3)} o18 : List i19 : betti(adjList_0) 0 1 o19 = total: 7 28 0: 7 28 o19 : BettiTally i20 : minimalBetti J 0 1 2 3 4 5 o20 = total: 1 12 25 21 10 3 0: 1 . . . . . 1: . 12 25 15 . . 2: . . . 6 10 3 o20 : BettiTally