NodalCurves -- construction of nodal rational curves

Description

What’s new: Minor corrections

Version 0.4.

Overview: Starting from a collection of g pairs of points P_i, Q_i the canonical image of the rational curves with nodes at these points is constructed. Similarly, embedding of these curves by the linear series K ⊗L with L an n-torsion bundle is constructed.

One main goal is to verify two conjectures. Conjecture A is the Prym-Green Conjecture on syzygies of Prym canonical curves of even genus g and level n, see Farkas, Ludwig [2008] for small values of g and level n. This works up to genus 16 with the exception of genus (g,n)=(8,2) or (16,2). The case of genus 16 requires a lot of time and memory.

Conjecture B, see Chiodo, Eisenbud, Farkas, Schreyer [2012] (HREF has to be corrected), concerns syzygies of torsion bundles L^k in Prym canonical space P(H⁰(C,K⊗L) again for curves of even genus.

Setup:

This package requires Macaulay2 version 1.4 or newer.

Install this Package by doing

installPackage("NodalCurves")

Examples:

Author

Frank-Olaf Schreyer <schreyer@math.uni-sb.de>

Version

This documentation describes version 0.4.2 of NodalCurves.

Source code

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

Exports

Functions and commands
- canonicalMultipliers -- compute the canonical multipliers of a rational curves with nodes
- criticalKoszulMatrixForPrymCurve -- compute the Koszul matrix for the critical Betti number in the Prym-Green conjecture
- criticalKoszulMatrixForTorsionBundle -- compute the Koszul matrix for the critical Betti number in the Prym-Green conjecture
- outputForSingular -- get a string for a matrix over a finite field
- randomCanonicalNodalCurve -- get a random canonical nodal curve of genus g
- randomPrymCanonicalNodalCurve -- get a random Prym canonical nodal curve of genus g, level n and k twisted multipliers
- syzygiesOfPrymCanonicalNodalCurve -- compute syzygies of a random Prym canonical nodal of genus g and level n
- syzygiesOfTorsionBundle -- compute syzygies of a Torsion bundles in a Prym canonical embedding for a g-nodal rational random example.
- verifyConjA -- verify the Prym-Green conjecture
- verifyConjB -- verify Conjecture B on the syzygies of torsions bundles in Prym canonical space
Symbols
- Printing -- ask for some printing
- Timings -- ask for some timing