According to Verra [Ve], a general genus 14 curve C arizes as the residual intersection of the 5 quadrics in the homogeneous ideal of a general normal curve E of genus 8 and degree 14 in ℙ6. These in turn can be constructed using Mukai’s Theorem on genus 8 curves: Every smooth genus 8 curve with general Clifford index arizes as the intersection of the Grassmannian G(2,6) ⊂ℙ14 with a transversal ℙ7. Taking ℙ7 as the span of general or random 8 points
gives E together with a general divisor H=KE+D1-D2 of degree 14 where D1=p1+...+p4 and D2=p5+...+p8.
The fact that the example below works can be seen as computer aided proof of the unirationality of M14. It proves the unirationality of M14 for fields of the choosen finite characteristic 10007, for fields of characteristic 0 by semi-continuity, and, hence, for all but finitely many primes p.
i1 : setRandomSeed("alpha"); |
i2 : FF=ZZ/10007; |
i3 : S=FF[x_0..x_6]; |
i4 : time I=randomCurveGenus14Degree18inP6(S); -- used 3.8282 seconds o4 : Ideal of S |
i5 : betti res I 0 1 2 3 4 5 o5 = total: 1 13 45 56 25 2 0: 1 . . . . . 1: . 5 . . . . 2: . 8 45 56 25 . 3: . . . . . 2 o5 : BettiTally |