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

Version

This documentation describes version 0.5 of RandomSpaceCurves.

Source code

The source code from which this documentation is derived is in the file RandomSpaceCurves.m2.

Exports

• Functions and commands
• Methods
• "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
• 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 .