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# canonicalSeries -- Canonical series of the normalization of a plane curve

## Synopsis

• Usage:
M = canonicalSeries R
• Inputs:
• R, a ring, ring of a plane curve
• Optional inputs:
• Conductor => ..., default value null
• Outputs:
• M, , a 1 x g matrix representing the canonical series

## Description

Computing the canonical linear series

 i1 : kk = QQ o1 = QQ o1 : Ring i2 : S = kk[x,y,z] o2 = S o2 : PolynomialRing i3 : C1 = ideal (y^3 - x^2*(x-z)) -- cubic with a node; geometric genus 0 3 3 2 o3 = ideal(- x + y + x z) o3 : Ideal of S i4 : C2 = ideal(x^2+y^2+z^2) --nonsingular conic 2 2 2 o4 = ideal(x + y + z ) o4 : Ideal of S i5 : C3 = ideal (x^4+y^4+z^4) -- smooth curve of genus 3 4 4 4 o5 = ideal(x + y + z ) o5 : Ideal of S i6 : canonicalSeries(S/C1) o6 = 0 / S \1 o6 : Matrix |---------------| <-- 0 | 3 3 2 | \- x + y + x z/ i7 : canonicalSeries(S/C2) o7 = ideal 0 S o7 : Ideal of ------------ 2 2 2 x + y + z i8 : canonicalSeries(S/C3) o8 = | x y z | / S \1 / S \3 o8 : Matrix |------------| <-- |------------| | 4 4 4| | 4 4 4| \x + y + z / \x + y + z /

## Ways to use canonicalSeries :

• canonicalSeries(Ideal)
• canonicalSeries(Ring)

## For the programmer

The object canonicalSeries is .