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NodalCurves :: verifyConjB

verifyConjB -- verify Conjecture B on the syzygies of torsions bundles in Prym canonical space

Synopsis

Description

Let (C, L) be a general curve of even genus g and L ∈  Pic0(C) an n-torsion bundle. In Chiodo, Eisenbud, Farkas, Schreyer, [2012] (HREF has to be corrected) it is conjectured that the syzygies of the module of global sections H0*(Pg-2,KC ⊗Lk) as an Sym H0(C,KC ⊗L) module are pure unless g ≡2 mod 4, binomial(g-3,g/2-1) ≡1 mod 2 and L is 2k-1 torsion. In the exceptional cases we conjecture precisely one extra syzygy.

With this function the conjecture can be verified for small g by computing the syzygies of a random g-nodal example over a finite ground field, and arguing by semi-continuity of betti numbers. The options allow to ask for printing and timing.

Printing=>true results in

1) printing of the naively expected betti table of the resolution of the torsion bundle

2) betti number of the critical Koszul cohomology matrix and over the ground field and the betti number of its syzygy matrix. Zero or one syzygy is the conjecture.

Timings=>true results in the priniting of three timings:

1) the time to compute the Koszul matrix as a matrix of polynomials

2) the time to convert this matrix to a matrix over the ground field

3) the time to compute the syzygies of this matrix.

Step 3 takes a lot of time, e.g. for g=16 is took us on our machine 20? days.
i1 : time verifyConjB(8,3,2,Printing=>true)
warning: clearing value of symbol t to allow access to subscripted variables based on it
       : debug with expression   debug 5504   or with command line option   --debug 5504
naively expected syzygies:
       0  1  2  3  4 5
total: 8 32 40 40 32 8
    0: 8 32 40  .  . .
    1: .  .  . 40 32 8
         0  1          0
(total: 70 70, total: 70)
    -1:  . 70      0: 70
     0: 70  .
     -- used 0.66943 seconds

o1 = true
i2 : time apply(3,i->verifyConjB(6,5,i+2))
     -- used 0.417772 seconds

o2 = {true, true, true}

o2 : List
i3 : time verifyConjB(10,3,2,Printing=>true,Timings=>true)
warning: clearing value of symbol t to allow access to subscripted variables based on it
       : debug with expression   debug 5504   or with command line option   --debug 5504
naively expected syzygies:
        0  1   2   3   4   5  6  7
total: 10 60 140 140 140 140 60 10
    0: 10 60 140 140   .   .  .  .
    1:  .  .   .   . 140 140 60 10
     -- used 3.01152 seconds
     -- used 0.030282 seconds
     -- used 3.92035 seconds
          0   1           0 1
(total: 315 315, total: 315 1)
    -1:   . 315     -1:   . 1
     0: 315   .      0: 315 .
     -- used 7.19932 seconds

o3 = true

Ways to use verifyConjB :