randomCurveGenus14Degree18inP6 -- Compute a random curve of genus 14 of Degree 18 in \PP^6

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 \PP^6. 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 $G(2,6) \subset \PP^{14}$ with a transversal $\PP^7$. Taking $\PP^7$ as the span of general or random $8$ points $$p_1,\ldots, p_8 \in{} G(2,6)$$ gives $E$ together with a general divisor $ H=K_E+D_1-D_2$ of degree 14 where $D_1=p_1+\ldots+p_4$ and $D_2=p_5+\ldots+p_8$.

The fact that the example below works can be seen as computer aided proof of the unirationality of $M_{14}$. It proves the unirationality of $M_{14}$ for fields of the chosen 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 1.16203 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