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

Version

This documentation describes version 0.5 of RandomSpaceCurves.

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/master/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 RandomSpaceCurves.m2:786:0.