RandomCurves -- constructing random curves in various ways

Description

A RandomObject in this sense is a hashTable consisting of two functions, one that can construct a random point and the other that can certify that the thing constructed has the desired property, say being a smooth space curve of the desired genus and degree. This is employed through a call such as random spaceCurve, which returns the constructor function; thus to get a random curve of degree d and genus g one must do something like

(random spaceCurve)(d,g,R)

where R is the homogeneous coordinate ring of P^3.

For a different approach, see the package SpaceCurves; there special curves in P^3 of every genus and degree that is allowed by Castelnuovo's theorem are constructed on surfaces of degree <=4, following the Theorem of Gruson and Peskine.

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

This package also generates random nodal plane curves and provides related methods.

Also 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.

Also This package bundles the constructions for random points in the moduli spaces of curves $M_g$ for $g \leq 14$ based on the proofs of unirationality of $M_g$ by Severi, Sernesi, Chang-Ran and Verra.

Further, it provides for random canonical curves made from nodal plane curves.

For random smooth curves defined over very small fields, see the package RandomCurvesOverVerySmallFiniteFields

Authors

Version

This documentation describes version 1.0 of RandomCurves, released March 1, 2011, updated and combined on June 8, 2026 by D Eisenbud.

Citation

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

@misc{RandomCurvesSource,
  title = {{RandomCurves: random smooth curves up to genus 14. Version~1.0}},
  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
- completeLinearSystemOnNodalPlaneCurve -- Compute the complete linear system of a divisor on a nodal plane curve
- expectedBetti -- compute the expected betti table from the Hilbert numerator
- hilbertNumerator -- calculate Hilbert numerator from Hilbert function
- imageUnderRationalMap -- Compute the image of the scheme under a rational map
- knownUnirationalComponentOfSpaceCurves -- check whether there is a unirational construction for a component of the Hilbert scheme of space curves
- 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
- certifyHartshorneRaoModule(Module,ZZ,List,PolynomialRing) (missing documentation)
- certifyRandomSpaceCurve(Ideal,ZZ,ZZ,PolynomialRing) (missing documentation)
- completeLinearSystemOnNodalPlaneCurve(Ideal,List) -- see completeLinearSystemOnNodalPlaneCurve -- Compute the complete linear system of a divisor on a nodal plane curve
- 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
- imageUnderRationalMap(Ideal,Matrix) -- see imageUnderRationalMap -- Compute the image of the scheme under a rational map
- knownUnirationalComponentOfSpaceCurves(ZZ,ZZ) -- see knownUnirationalComponentOfSpaceCurves -- check whether there is a unirational construction for a component of the Hilbert scheme of space curves
- 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
- randomSpaceCurve(ZZ,ZZ,PolynomialRing) (missing documentation)
Other things
- canonicalCurve -- Compute a random canonical curve of genus less or equal to 14
- canonicalCurveGenus14 -- compute a random curve of genus 14 in its canonical embedding
- curveGenus14Degree18inP6 -- compute a random curve of genus 14 and degree 18 in $\mathbb{P}^6$
- distinctPlanePoints -- Generates the ideal of k random points in the coordinate ring $R$ of $\\P^{ 2}$
- hartshorneRaoModule -- Compute a random Hartshorne-Rao module
- nodalPlaneCurve -- get a random nodal plane curve
- spaceCurve -- constructs a RandomObject that can be used to construct a space curve.

For the programmer

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

The source of this document is in /build/reproducible-path/macaulay2-1.26.06+ds/M2/Macaulay2/packages/RandomCurves.m2:891:0.