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

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 /