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.
i1 : time fM=syzygiesOfTorsionBundle(8,3,2,Printing=>true);
naively expected syzygies:
0 1 2 3 4 5
total: 7 28 35 35 28 7
0: 7 28 35 . . .
1: . . . 35 28 7
linear strand in the example:
0 1 2
total: 7 28 35
0: 7 28 35
-- used 0.663584 seconds
|
i2 : time fM=syzygiesOfTorsionBundle(6,3,2,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
total: 5 10 10 5
0: 5 10 . .
1: . . 10 5
linear strand in the example:
0 1 2
total: 5 10 1
0: 5 10 1
-- used 0.125642 seconds
|
i3 : time fM=syzygiesOfTorsionBundle(6,5,2,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
total: 5 10 10 5
0: 5 10 . .
1: . . 10 5
linear strand in the example:
0 1
total: 5 10
0: 5 10
-- used 0.185291 seconds
|
i4 : time fM=syzygiesOfTorsionBundle(6,5,3,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
total: 5 10 10 5
0: 5 10 . .
1: . . 10 5
linear strand in the example:
0 1 2
total: 5 10 1
0: 5 10 1
-- used 0.149836 seconds
|