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kGonalNodalCurves :: kGonalNodalCurves

kGonalNodalCurves -- consrtuction of rational k-gonal g-nodal canonical curves


We construct two sets of g points in PP^1, represented by two 2xg matrices P, and Q in such a way that the curve constructed by gluing this points to nodes as implemented in the the Macaulay2 package NodalCurves gives a nodal curve that carries a g^1_k.

The variety swept out by a g^1_k on a canonical curve is a (k-1) dimensional rational normal scroll X We want to check whether the Betti numbers beta_{m+c,m+c+1}(C)coincide with those of the scroll for c>=0 and m=ceiling((g-1)/2). This package also provides several functions which can be handy for the computation of the critical Betti number beta_{m,m+1}(C)using Koszul cohomolgy.See Koszul Cohomology and the Geometry of Projective Varieties for a good survey on the topic.

Construction of rational d-gonal g-nodal canonical curves

  • getFieldAndRing -- computes a prime number p>=n, the field kk=ZZ/p and the coordinate Ring S of PP^1(kk)
  • getFactors -- computes linear and quadratic factors of a polynomial
  • getCoordinates -- computes vanishing loci of a linear polynomial in two variables
  • pickGoodPoints -- computes g pairs of points and gluing data between them
  • listOfFactors -- finds a map f: S^2(-k)->S such that det(f|pt) factors factors for sufficiently many points pt in PP^1
  • linFactorsToQuadric -- computes quadrics needed to construct the matrix defining the canonical embedding
  • getSections -- computes a matrix defining the canonical embedding
  • sectionsFromPoints -- computes matrix defining the canonical embedding from the 2g points
  • canonicalMultipliers -- compute the canonical multipliers of a rational curves with nodes
  • lineBundleFromPointsAndMultipliers -- computes basis of a line bundle from the 2g points P_i, Q_i and the multipliers
  • idealOfNodalCurve -- computes the ideal I of a k-gonal g-nodal canonical curve.
  • idealOfNodalCurveByPoints -- computes ideal of a canonical nodal curve constructed from 2g points
  • scrollType -- determines the type of the scroll

Computation of the critical Betti number


This package requires Macaulay2 Version 1.9 or newer



This documentation describes version 0.2 of kGonalNodalCurves.

Source code

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