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Algebraic Curves
New Features:
- A package of functions to compute smallest degree functions/maps to
as well as smallest degree birational plane models for low genus curves has been added.
These are geometrically smallest degree, meaning they may be constructed over
a finite extension of the base field in some cases. The relevant intrinsics are
summarised in the next four items.
- CliffordIndexOne. For a trigonal curve of any genus
3
, computes an
explicit degree 3 map to
. In the exceptional Clifford index one
where the curve has genus 6 and gonality 4, computes an explicit birational
map to a non-singular plane quintic. This uses Lie algebras via the algorithm
of Schicho and Sevilla.
- GenusNGonalMap - here N can be 2 to 6. Computes an explicit
gonal (smallest degree) map to
for an algebraic curve of genus
2,3,4,5 or 6. For the hyperelliptic cases, uses the existing code; for the
Clifford index one cases uses the previous intrinsic and for the genus 5
and 6 gonality 4 cases uses an algorithm of Harrison.
- Genus5PlaneCurveModel and Genus6PlaneCurveModel. Computes
smallest degree (5 or 6) birational plane models of genus 5 or 6
curves that are not hyperelliptic or double covers of a genus 1 curve.
Uses a variant of the method used to compute gonal maps.
- The new intrinsic IntersectionNumbers uses an algorithm of Jan Hilmar
and Chris Smyth to compute and return all intersection places along with the
corresponding local intersection multiplicity for two plane projective curves
over Q in a single computation. (To be included in an early patch release).
- The intrinsic Completion applied to a function field of a curve at a
place of the curve can now be computed directly. (V2.18-8)
Changes:
- Expand can now be applied to places of degree greater than 1. (V2.18-8)
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