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RandomGenus14Curves :: randomCurveGenus14Degree18inP6

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

Synopsis

Description

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 7. Taking 7 as the span of general or random 8 points

p1,..., p8 ∈ G(2,6)

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 2.25996 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

Ways to use randomCurveGenus14Degree18inP6 :

  • randomCurveGenus14Degree18inP6(PolynomialRing)