randomPrymCanonicalCurveOfGenus7 -- pick a random Prym canonical curve of genus 7

Acoording to Farkas, Verra [2012] the moduli space R₇ of Prym canonical curves of genus 7 is unirational. The proof is bassed on the fact that a general Prym canonical embedded curve C of genus 7 lies on a unique Nikulin surface X, the intersection of the three quadrics in the homogeneous ideal of C. If L₁, ..., L₈ denote the 8 lines on the Nikulin surface then the linear system |C-(L₁+...+L₇)| is zero dimensional and consists of a rational normal curve C5 which has L₁, ..., L₇ as secants lines. The unirational construction reverses this observation. The union C5 ∩L₁ ∩...∩L₇ ⊂ℙ⁵ of a rational normal curve of degree 5 with seven general secant lines is contained in a unique Nikulin surface X, and a general C ∈|R5+L₁+...+L₇| on X gives the desired general C ∈R₇.

i1 : p=10007,kk=ZZ/p,R=kk[x_0..x_5]

o1 = (10007, kk, R)

o1 : Sequence

i2 : time I=randomPrymCanonicalCurveOfGenus7(R);
     -- used 0.572635 seconds

o2 : Ideal of R

i3 : (dim I,degree I, genus I)

o3 = (2, 12, 7)

o3 : Sequence

i4 : betti res I

            0  1  2  3 4
o4 = total: 1 11 27 24 7
         0: 1  .  .  . .
         1: .  3  .  . .
         2: .  8 27 24 7

o4 : BettiTally