PrymCanonicalCurves -- construction of Prym canonical curves

Description

Overview: We randomly construct smooth Prym canonical curves of genus 7 over a finite prime field based on the unirationality proof of Farkas, Verra [2012]. We then add a node to these curve to obtain a random 1-nodal Prym canonical curve of genus 8. We also have a construction of smooth Prym canonical curve of genis 8, which are not general in their moduli space, because they have a plane model of degree 7. For more see "getCanonicalCurveOfGenus8With2Torsion". Starting from these curves we construct Prym canonical curves of higher genus by adding nodes. The main goal is to verify the Prym Green conjecture Farkas, Ludwig [2008] for small values of g.

The conjecture says that a general Prym canonical curve of even genus has a pure resolution. In case of genus g=8 the verification fails: We always find one extra syzygy. In the range g=10..14 the verification works, g=16 gives again one extra syzygy. Experiments for the last cases are however expensive in cpu time and memory.

Finally we study the extra syzygy in genus 8. For more on this, see "extraSyzygyInGenus8".

Setup:

This package requires Macaulay2 version 1.4 or newer.

Install this Package by doing

installPackage("PrymCanonicalCureves")

Author

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

Version

This documentation describes version 0.3 of PrymCanonicalCurves.

Source code

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

Exports

Functions and commands
- addANodeAndExtend2Torsion -- add a node to a canonical curve and extend a 2-Torsion divisor to the new curve
- check2Torsion -- Check that D_0-D_1 represents is a two torsion divisor on the curve defined by J
- extraSyzygyInGenus8 -- study the extra syzygy of a random Prym canonical curve of genus 8
- getAPrymCanonicalCurve -- get a Prym canonical curve
- getCanonicalCurveOfGenus8With2Torsion -- Construct a canonical curve C of genus 8 together with a 2-torsion divisor and n extra points
- getSepticOfGenus8With2Torsion -- Construct a plane septic curve C of geometric genus 8 together with a 2-torsion divisor
- prymCanonicalEmbedding -- Compute the Prym canonical image
- randomOneNodalPrymCanonicalCurveOfGenus8 -- pick a random Prym canonical curve of arithmetic genus 8 with one node
- randomPrymCanonicalCurveOfGenus7 -- pick a random Prym canonical curve of genus 7
- simplifyDivisor -- Simplify the representation of a divisor on a curve.