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# RandomSpaceCurves -- Construction of random space curves of various kinds.

## Description

This package provides the construction of random curves $C \subset \mathbb{P}^{ 3}$ for various values for its degree $d$ and genus $g$. A space curve $C \subset \mathbb{P}^{ 3}$ is constructed via its Hartshorne-Rao module $M= H^1_*(\mathcal{I}_C(n))$. In particular, there are constructions for random points in $M_g$ for $g=11,12,13$.

For a algorithms and theoretical background see Needles in a Haystack

## Authors

- Hans-Christian Graf v. Bothmer <bothmer@uni-math.gwdg.de>
- Florian Geiss <fg@math.uni-sb.de>
- Frank-Olaf Schreyer <schreyer@math.uni-sb.de>

## Version

This documentation describes version 0.5 of RandomSpaceCurves, released March 1, 2011.

## Citation

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

    @misc{RandomSpaceCurvesSource,
      title = {{RandomSpaceCurves: random smooth space curves. Version~0.5}},
      author = {Hans-Christian Graf v. Bothmer and Florian Geiss and Frank-Olaf Schreyer},
      howpublished = {A \emph{Macaulay2} package available at
        \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
    }

## Exports

- Functions and commands
  - expectedBetti -- compute the expected betti table from the Hilbert numerator
  - hilbertNumerator -- calculate Hilbert numerator from Hilbert function
  - knownUnirationalComponentOfSpaceCurves -- check whether there is a unirational construction for a component of the Hilbert scheme of space curves
- Methods
  - certifyHartshorneRaoModule(Module,ZZ,List,PolynomialRing) (missing documentation)
  - certifyRandomSpaceCurve(Ideal,ZZ,ZZ,PolynomialRing) (missing documentation)
  - constructHartshorneRaoModule(ZZ,List,PolynomialRing) (missing documentation)
  - `expectedBetti(RingElement)` -- see expectedBetti -- compute the expected betti table from the Hilbert numerator
  - expectedBetti(List,ZZ) -- compute the expected betti table from the Hilbert numerator
  - expectedBetti(ZZ,ZZ,ZZ) -- compute the expected betti table from the Hilbert numerator
  - `hilbertNumerator(List,ZZ,RingElement)` -- see hilbertNumerator -- calculate Hilbert numerator from Hilbert function
  - `knownUnirationalComponentOfSpaceCurves(ZZ,ZZ)` -- see knownUnirationalComponentOfSpaceCurves -- check whether there is a unirational construction for a component of the Hilbert scheme of space curves
  - randomSpaceCurve(ZZ,ZZ,PolynomialRing) (missing documentation)
- Other things
  - hartshorneRaoModule -- Compute a random Hartshorne-Rao module
  - spaceCurve -- Generates the ideal of a random space curve of genus g and degree d

## For the programmer

The object RandomSpaceCurves is a package, defined in RandomSpaceCurves.m2.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/RandomSpaceCurves.m2:786:0.