Is the section df associated to a rational function f on a curve X a...
Is the section df associated to a rational function f on a curve X a global section of the canonical sheaf ? 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 , which one could write rigorously as .
Beware however that not all rational sections of are of this form: the simplest example is on (or on ) 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 :
For example if on , then , which is not a section of since it has a pole worse than had f!
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