Is the section df associated to a rational function f



Answered question


Is the section df associated to a rational function f on a curve X a global section of the canonical sheaf ωX? I know its zeroes are the ramification points, but does it have poles?

Answer & Explanation



Beginner2022-02-16Added 6 answers

Yes, df is a rational section of ωX, which one could write rigorously as dfϵΓ(X,ωXOXKX).
Beware however that not all rational sections of ωX are of this form: the simplest example is dzz on C(or on PP{C}1) which is not the differential of any rational function.


Beginner2022-02-17Added 9 answers

Of course if f is not regular (i.e. if f has poles) df will not be a global section of ωX:
dfϵΓ(X,ωXOXKX) Γ(X,ωX)
For example if f=1z on X=C, then df=1z2dz, which is not a section of ωX since it has a pole worse than had f!

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