H0(C,KC⊗Lk) ⊕H0(C,KC2⊗Lk+1) ⊕H0(C,KC3⊗Lk+3)) ⊕...
an artinian reduction by sg-3,sg-2 has shapeA=H0(C,KC ⊗Lk) ⊕H0(C,KC2 ⊗Lk+1) / H0(C,KC⊗Lk)<sg-3,sg-2>
The desired Koszul matrix is koszul(m-1,<s0,...,sg-4>) ⊗sB with values in A2. This is a binomial(g-3,m-2) x (g-1)*binomial(g-3,m-1) matrix of binary forms of degree 4g-4 with values in a g-1 dimensional complementary subspace ofH0(C,KC⊗Lk)<sg-3,sg-2> ⊂H0(C,KC2 ⊗Lk+1) ⊂K[x0,x1]4g-4
i1 : (g,n,k)=(8,3,2) o1 = (8, 3, 2) o1 : Sequence |
i2 : M=criticalKoszulMatrixForTorsionBundle(g,n,k); ZZ 10 ZZ 70 o2 : Matrix (-----[x , x ]) <--- (-----[x , x ]) 10303 0 1 10303 0 1 |
i3 : m=lift(g/2,ZZ) o3 = 4 |
i4 : (rank target M, rank source M) == (binomial(g-3,m-2), (g-1)*binomial(g-3,m-1)) o4 = true |
i5 : isHomogeneous M o5 = true |
i6 : rank source mingens ideal M == g-1 o6 = true |