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NodalCurves :: criticalKoszulMatrixForTorsionBundle

criticalKoszulMatrixForTorsionBundle -- compute the Koszul matrix for the critical Betti number in the Prym-Green conjecture

Synopsis

Description

First an n-Prym canonical series s=H0(C,KC⊗L) of a g-nodal rational curve and the serie sB=H0(C,KC ⊗Lk) is computed. Then an artinian reduction of the critical Kosul matrix is computed: Since module of the torsion bundle

H0(C,KC⊗Lk) ⊕H0(C,KC2⊗Lk+1) ⊕H0(C,KC3⊗Lk+3)) ⊕...

an artinian reduction by sg-3,sg-2 has shape

A=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 of

H0(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

Ways to use criticalKoszulMatrixForTorsionBundle :