experiment1 -- count the number of curves with extra syzygies and their ranks

Synopsis

Usage:
I=experiment1(p,N)
Inputs:
- p, an integer, a prime number specifying a finite field
- N, an integer, number of curves in Pic¹4(M₈) we will test for extra syzygies
Outputs:
- a tally, of a list of N pairs (Betti table of ideal, rank of syzygy)

Description

We compute N random examples of normal curves of genus 8 and degree 14 in P⁶ and count how often extra syzygies occur and what the rank of the syzygy is. For a given prime p it is reasonable to take N=p².

i1 : kk=ZZ/7;R=kk[x_0..x_6];

i3 : setRandomSeed("quickly")

o3 = 121192488220970

i4 : time experiment1(7,3)
     -- used 20.5045 seconds

                   0 1
o4 = Tally{(total: 1 7, -1) => 2}
                0: 1 .
                1: . 7
                   0 1
           (total: 1 8, 6) => 1
                0: 1 .
                1: . 7
                2: . 1

o4 : Tally

Rank -1 means that, there is not precisely one extra syzygy.