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

- Usage:
    `randomCurveGenus14Degree18inP6 S`
- Inputs:
  - S, a polynomial ring, homogeneous coordinate ring of \PP^6
- Optional inputs:
  - Certify (missing documentation) => ..., default value false,
- Outputs:
  - an ideal, the ideal of a curve C of genus 14 and degree 18 in \PP^6

## 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 \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");
     -- setting random seed to 10206284518

    i2 : FF=ZZ/10007;

    i3 : S=FF[x_0..x_6];

    i4 : time I=randomCurveGenus14Degree18inP6(S);
     -- used 1.60983s (cpu); 1.38306s (thread); 0s (gc)

    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)`

## For the programmer

The object randomCurveGenus14Degree18inP6 is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/RandomGenus14Curves.m2:323:0.