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NodalCurves :: criticalKoszulMatrixForPrymCurve

criticalKoszulMatrixForPrymCurve -- 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 canonical series omega is computed. Then an artinian reduction of the critical Kosul matrix is computed: Since the dual module is

H0(C,KC) ⊕H0(C,KC2⊗L) ⊕H0(C,KC3⊗L2) ⊕...

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

A=H0(C,KC) ⊕H0(C,KC2 ⊗L) / H0(C,KC)<sg-3,sg-2> ⊕...

and Hilbert function (g,g-3,1) The desired Koszul matrix is koszul(m,<s0,...,sg-4>) ⊗sB with values in A2 which is returned. The result is a binomial(g-3,m-1) x g*binomial(g-3,m) matrix of binary forms of degree 4g-4 with values in a g-3 dimensional complementary subspace of

H0(C,KC)<sg-3,sg-2> ⊂H0(C,KC2 ⊗L) ⊂kk[x0,x1]4g-4

i1 : (g,n)=(8,2);
i2 : M=criticalKoszulMatrixForPrymCurve(g,n);

               ZZ          10         ZZ          40
o2 : Matrix (-----[x , x ])   <--- (-----[x , x ])
             12451  0   1           12451  0   1
i3 : isHomogeneous M

o3 = true
i4 : betti syz( M,DegreeLimit=>4*g-4)

             0 1
o4 = total: 40 1
        27:  . 1
        28: 40 .

o4 : BettiTally
i5 : m= lift(g/2,ZZ)

o5 = 4
i6 : (rank target M, rank source M) == (binomial(g-3, m-1), g*binomial(g-3,m))

o6 = true
i7 : rank source mingens ideal M == g-3

o7 = true

Ways to use criticalKoszulMatrixForPrymCurve :