This is RandomGenus14Curves.info, produced by Macaulay2, version 1.25.11

RandomGenus14Curves -- Construction of random curves of genus 14
****************************************************************

Description
===========

In this package the unirationality construction of the moduli space $M_{14}$ of
curves of genus 14 due to Verra is implemented. The main references are

\ \ \ \ \ [Mu] S. Mukai, Curves, $K3$ surfaces and Fano $3$-folds of genus
$\leq 10$. Algebraic geometry and commutative algebra, Vol. I, 357-377,
Kinokuniya, Tokyo, 1988.

\ \ \ \ \ [Ve] A. Verra, The unirationality of the moduli spaces of curves of
genus 14 or lower. Compos. Math. 141 (2005), no. 6, 1425-1444.

Authors
=======

  * Frank-Olaf Schreyer (see http://www.math.uni-sb.de/ag/schreyer/ )
    <schreyer@math.uni-sb.de>
  * Hans-Christian Graf v. Bothmer (see http://www.crcg.de/wiki/User:Bothmer )
    <bothmer@uni-math.gwdg.de>

Version
=======

This documentation describes version 0.6 of RandomGenus14Curves, released March
4, 2011.

Citation
========

If you have used this package in your research, please cite it as follows:

+-----------------------------------------------------------------------------+
|@misc{RandomGenus14CurvesSource,                                             |
|  title = {{RandomGenus14Curves: random smooth curves of genus 14. Version   |
|  author = {Frank-Olaf Schreyer and Hans-Christian Graf v. Bothmer},         |
|  howpublished = {A \emph{Macaulay2} package available at                    |
|    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages   |
|}                                                                            |
|-----------------------------------------------------------------------------|
|~0.6}},                                                                      |
|                                                                             |
|                                                                             |
|}}                                                                           |
+-----------------------------------------------------------------------------+

Exports
=======

  * Functions and commands
      * randomCanonicalCurveGenus8with8Points
        -- Compute a random canonical
        curve of genus 8 with 8 marked point
      * randomCurveGenus8Degree14inP6, --
        Compute a random normal curve of genus g=8 and degree 14 in \PP^6
      * randomCurveGenus14Degree18inP6,
        -- Compute a random curve of genus 14 of Degree 18 in \PP^6
  * Methods
      * "randomCanonicalCurveGenus8with8Points(PolynomialRing)" -- see
        randomCanonicalCurveGenus8with8Points,
        -- Compute a random canonical
        curve of genus 8 with 8 marked point
      * "randomCurveGenus8Degree14inP6(PolynomialRing)" -- see
        randomCurveGenus8Degree14inP6, --
        Compute a random normal curve of genus g=8 and degree 14 in \PP^6
      * "randomCurveGenus14Degree18inP6(PolynomialRing)" -- see
        randomCurveGenus14Degree18inP6, --
        Compute a random curve of genus 14 of Degree 18 in \PP^6
  * Other things
      * canonicalCurveGenus14 -- compute a random
        curve of genus 14 in its canonical embedding
      * curveGenus14Degree18inP6 -- compute a
        random curve of genus 14 and degree 18 in $\mathbb{P}^6$

For the programmer
==================

The object RandomGenus14Curves, is a package (Macaulay2Doc),
defined in RandomGenus14Curves.m2.

-------------------------------------------------------------------------------

The source of this document is in RandomGenus14Curves.m2:180:0.

canonicalCurveGenus14 -- compute a random curve of genus 14 in its canonical embedding
**************************************************************************************

  * Usage:
        (random canonicalCurveGenus14)(R)
  * Inputs:
      * R, a polynomial ring (Macaulay2Doc), coordinate
        ring of $\mathbb{P}^13$
  * Outputs:
      * an ideal (Macaulay2Doc), in R, ideal of the canonical
        curve

Description
===========

+-----------------------------------------+
|i1 : setRandomSeed("alpha");             |
| -- setting random seed to 10206284518   |
+-----------------------------------------+
|i2 : R=ZZ/101[x_0..x_13];                |
+-----------------------------------------+
|i3 : C=(random canonicalCurveGenus14)(R);|
|                                         |
|o3 : Ideal of R                          |
+-----------------------------------------+
|i4 : (dim C, degree C, genus C)          |
|                                         |
|o4 = (2, 26, 14)                         |
|                                         |
|o4 : Sequence                            |
+-----------------------------------------+

For the programmer
==================

The object canonicalCurveGenus14, is an instance
of the type RandomObject (RandomObjects).

-------------------------------------------------------------------------------

The source of this document is in RandomGenus14Curves.m2:201:0.

curveGenus14Degree18inP6 -- compute a random curve of genus 14 and degree 18 in $\mathbb{P}^6$
**********************************************************************************************

  * Usage:
        (random curveGenus14Degree18inP6)(R)
  * Inputs:
      * R, a polynomial ring (Macaulay2Doc), coordinate
        ring of $\PP^6$
  * Outputs:
      * an ideal (Macaulay2Doc), in R, ideal of the curve

Description
===========

+--------------------------------------------+
|i1 : setRandomSeed("alpha");                |
| -- setting random seed to 10206284518      |
+--------------------------------------------+
|i2 : R=ZZ/101[x_0..x_6];                    |
+--------------------------------------------+
|i3 : C=(random curveGenus14Degree18inP6)(R);|
|                                            |
|o3 : Ideal of R                             |
+--------------------------------------------+
|i4 : (dim C, degree C, genus C)             |
|                                            |
|o4 = (2, 18, 14)                            |
|                                            |
|o4 : Sequence                               |
+--------------------------------------------+

For the programmer
==================

The object curveGenus14Degree18inP6, is an
instance of the type RandomObject (RandomObjects).

-------------------------------------------------------------------------------

The source of this document is in RandomGenus14Curves.m2:223:0.

randomCanonicalCurveGenus8with8Points -- Compute a random canonical curve of genus 8 with 8 marked point
********************************************************************************************************

  * Usage:
        (I,idealsOfPts)=randomCanonicalCurveGenus8with8Points S
  * Inputs:
      * S, a polynomial ring (Macaulay2Doc), homogeneous
        coordinate ring of $\PP^7$
  * Outputs:
      * I, an ideal (Macaulay2Doc), a canonical curve C of genus 8
      * idealsOfPts, a list (Macaulay2Doc), 8 ideals of K-rational
        points on C

Description
===========

According to Mukai [Mu] any smooth curve of genus 8 and Clifford index 3 is the
transversal intersection $C=\PP^7 \cap\ G(2,6) \subset \ \PP^{15}$. In
particular this is true for the general curve of genus 8. Picking 8 points in
the Grassmannian $G(2,6)$ at random and \PP^7 as their span gives the result.

+-----------------------------------------------------------------------------+
|i1 : setRandomSeed("alpha");                                                 |
| -- setting random seed to 10206284518                                       |
+-----------------------------------------------------------------------------+
|i2 : FF=ZZ/10007;                                                            |
+-----------------------------------------------------------------------------+
|i3 : S=FF[x_0..x_7];                                                         |
+-----------------------------------------------------------------------------+
|i4 : (I,points)=randomCanonicalCurveGenus8with8Points S;                     |
+-----------------------------------------------------------------------------+
|i5 : betti res I                                                             |
|                                                                             |
|            0  1  2  3  4  5 6                                               |
|o5 = total: 1 15 35 42 35 15 1                                               |
|         0: 1  .  .  .  .  . .                                               |
|         1: . 15 35 21  .  . .                                               |
|         2: .  .  . 21 35 15 .                                               |
|         3: .  .  .  .  .  . 1                                               |
|                                                                             |
|o5 : BettiTally                                                              |
+-----------------------------------------------------------------------------+
|i6 : points                                                                  |
|                                                                             |
|o6 = {ideal (x  - 1611x , x  + 2414x , x  - 1550x , x  - 4251x , x  + 4315x ,|
|              6        7   5        7   4        7   3        7   2        7 |
|     ------------------------------------------------------------------------|
|     x  + 2701x , x  - 4090x ), ideal (x  + 1668x , x  - 2841x , x  - 2811x ,|
|      1        7   0        7           6        7   5        7   4        7 |
|     ------------------------------------------------------------------------|
|     x  - 2503x , x  - 2868x , x  + 3683x , x  + 4539x ), ideal (x  - 1972x ,|
|      3        7   2        7   1        7   0        7           6        7 |
|     ------------------------------------------------------------------------|
|     x  - 3164x , x  + 2439x , x  - 3731x , x  - 3223x , x  - 4255x , x  -   |
|      5        7   4        7   3        7   2        7   1        7   0     |
|     ------------------------------------------------------------------------|
|     3245x ), ideal (x  - 3018x , x  - 4221x , x  + 3745x , x  - 3558x , x  +|
|          7           6        7   5        7   4        7   3        7   2  |
|     ------------------------------------------------------------------------|
|     3717x , x  - 879x , x  - 4286x ), ideal (x  - 3067x , x  - 941x , x  +  |
|          7   1       7   0        7           6        7   5       7   4    |
|     ------------------------------------------------------------------------|
|     4350x , x  - 49x , x  + 4179x , x  - 3962x , x  + 2213x ), ideal (x  +  |
|          7   3      7   2        7   1        7   0        7           6    |
|     ------------------------------------------------------------------------|
|     4309x , x  + 2155x , x  - 613x , x  - 155x , x  - 542x , x  + 1237x , x |
|          7   5        7   4       7   3       7   2       7   1        7   0|
|     ------------------------------------------------------------------------|
|     + 2446x ), ideal (x  - 4950x , x  - 597x , x  - 4613x , x  - 1907x , x  |
|            7           6        7   5       7   4        7   3        7   2 |
|     ------------------------------------------------------------------------|
|     + 4217x , x  - 886x , x  + 3338x ), ideal (x  + 2599x , x  + 3310x , x  |
|            7   1       7   0        7           6        7   5        7   4 |
|     ------------------------------------------------------------------------|
|     + 1584x , x  + 3379x , x  + 1721x , x  - 3470x , x  + 3687x )}          |
|            7   3        7   2        7   1        7   0        7            |
|                                                                             |
|o6 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use randomCanonicalCurveGenus8with8Points:
==================================================

  * "randomCanonicalCurveGenus8with8Points(PolynomialRing)"

For the programmer
==================

The object randomCanonicalCurveGenus8with8Points,
is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in RandomGenus14Curves.m2:256:0.

randomCurveGenus8Degree14inP6 -- Compute a random normal curve of genus g=8 and degree 14 in \PP^6
**************************************************************************************************

  * Usage:
        I=randomCurveGenus8Degree14inP6 S
  * Inputs:
      * S, a polynomial ring (Macaulay2Doc), in 7
        variables
  * Outputs:
      * I, an ideal (Macaulay2Doc), of a curve of geometric genus
        8 and degree 14 in \PP^6

Description
===========

The construction is based on Mukai's unirational description of $M_{8,8}$ of
the moduli space of genus 8 with 8 marked points (see [Mu]).

+---------------------------------------+
|i1 : setRandomSeed("alpha");           |
| -- setting random seed to 10206284518 |
+---------------------------------------+
|i2 : FF=ZZ/10007;                      |
+---------------------------------------+
|i3 : S=FF[x_0..x_6];                   |
+---------------------------------------+
|i4 : I=randomCurveGenus8Degree14inP6 S;|
|                                       |
|o4 : Ideal of S                        |
+---------------------------------------+
|i5 : betti res I                       |
|                                       |
|            0 1  2  3  4 5             |
|o5 = total: 1 7 35 56 35 8             |
|         0: 1 .  .  .  . .             |
|         1: . 7  .  .  . .             |
|         2: . . 35 56 35 8             |
|                                       |
|o5 : BettiTally                        |
+---------------------------------------+

Ways to use randomCurveGenus8Degree14inP6:
==========================================

  * "randomCurveGenus8Degree14inP6(PolynomialRing)"

For the programmer
==================

The object randomCurveGenus8Degree14inP6,
is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in RandomGenus14Curves.m2:283:0.

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

  * Usage:
        randomCurveGenus14Degree18inP6 S
  * Inputs:
      * S, a polynomial ring (Macaulay2Doc), homogeneous
        coordinate ring of \PP^6
  * Optional inputs (Macaulay2Doc),:
      * Certify (missing documentation) => ..., default value false,
  * Outputs:
      * an ideal (Macaulay2Doc), 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 2.00766s (cpu); 1.76975s (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 (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in RandomGenus14Curves.m2:323:0.