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?

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bedevijuo3e · Accepted Answer

Yes, df is a rational section of ωX, which one could write rigorously as dfϵΓ(X,ωX⊗OXKX).    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.

an2gi2m9gg · Accepted Answer

Edit:  Of course if f is not regular (i.e. if f has poles) df will not be a global section of ωX:    dfϵΓ(X,ωX⊗OXKX) 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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