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# PlaneCurveLinearSeries -- Linear series on the normalization of a plane curve

## Description

This package implements procedures described in chapters 4, 5, and 14 of the book "The Practice of Curves", by David Eisenbud and Joe Harris.

If C is a (possibly singular) irreducible plane curve, it is possible to compute the complete linear series of a given divisor on the normalization C' of C by computations using data from the plane curve together with the conductor ideal $ann_C(C/C)$, which can be computed by Macaulay or supplied by the user.

The main routine of the package is linearSeries. If D0' and Dinf' are effective divisors on C' whose ideals, as schemes, are pulled back from ideals D0 and Dinf of C, then

ell = linearSeries(D0,Dinf)

returns a one-row matrix ell whose entries span a linear series with fixed point locus B on C' (including the conductor scheme) and form a basis of |D0'-Dinf'|+B.

The routine projectiveImage provides the image of the map to projective space given by |D0-Dinf|. There are special routines for the most important case, canonicalSeries and canonicalImage.

The functions addition and negative implement the group law in the case of a curve of genus 1.

## Version

This documentation describes version 1.0 of PlaneCurveLinearSeries.

## Source code

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

## Exports

• Functions and commands
• addition -- addition of smooth points on a curve of genus 1
• canonicalImage -- canonical model of the normalization of a plane curve
• canonicalSeries -- Canonical series of the normalization of a plane curve
• fromCoordinates -- Compute the ideal of a point from its coordinates
• geometricGenus -- Geometric genus of a (singular) plane curve
• linearSeries -- compute a linear series
• negative -- implements the inverse in the group law of a curve of genus 1
• projectiveImage -- Projective image of the map defined by a divisor or matrix
• toCoordinates -- coordinates of a point from its ideal
• Methods
• canonicalImage(Ideal) -- see canonicalImage -- canonical model of the normalization of a plane curve
• canonicalImage(Ring) -- see canonicalImage -- canonical model of the normalization of a plane curve
• canonicalSeries(Ideal) -- see canonicalSeries -- Canonical series of the normalization of a plane curve
• canonicalSeries(Ring) -- see canonicalSeries -- Canonical series of the normalization of a plane curve
• fromCoordinates(List,Ring) -- see fromCoordinates -- Compute the ideal of a point from its coordinates
• fromCoordinates(ZZ,ZZ,ZZ,Ring) -- see fromCoordinates -- Compute the ideal of a point from its coordinates
• geometricGenus(Ideal) -- see geometricGenus -- Geometric genus of a (singular) plane curve
• geometricGenus(Ring) -- see geometricGenus -- Geometric genus of a (singular) plane curve
• linearSeries(Ideal) -- see linearSeries -- compute a linear series
• linearSeries(Ideal,Ideal) -- see linearSeries -- compute a linear series
• linearSeries(List,List,Ring) -- see linearSeries -- compute a linear series
• linearSeries(List,Ring) -- see linearSeries -- compute a linear series
• negative(Ideal,Ideal) -- see negative -- implements the inverse in the group law of a curve of genus 1
• negative(List,List,Ring) -- see negative -- implements the inverse in the group law of a curve of genus 1
• projectiveImage(Ideal) -- see projectiveImage -- Projective image of the map defined by a divisor or matrix
• projectiveImage(Ideal,Ideal) -- see projectiveImage -- Projective image of the map defined by a divisor or matrix
• projectiveImage(List,List,Ring) -- see projectiveImage -- Projective image of the map defined by a divisor or matrix
• projectiveImage(List,Ring) -- see projectiveImage -- Projective image of the map defined by a divisor or matrix
• projectiveImage(Matrix) -- see projectiveImage -- Projective image of the map defined by a divisor or matrix
• toCoordinates(Ideal) -- see toCoordinates -- coordinates of a point from its ideal
• Symbols

## For the programmer

The object PlaneCurveLinearSeries is .