iristh3virusoo2

2022-02-15

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

bedevijuo3e

Yes, df is a rational section of ${\omega }_{X}$, which one could write rigorously as $dfϵ\mathrm{\Gamma }\left(X,{\omega }_{X}{\otimes }_{{O}_{X}}{K}_{X}\right)$.
Beware however that not all rational sections of ${\omega }_{X}$ are of this form: the simplest example is $\frac{dz}{z}$ on $\mathbb{C}$(or on $P{P}_{\left\{\mathbb{C}\right\}}^{1}$) which is not the differential of any rational function.

an2gi2m9gg

Edit:
Of course if f is not regular (i.e. if f has poles) df will not be a global section of ${\omega }_{X}$:

For example if $f=\frac{1}{z}$ on $X=\mathbb{C}$, then $df=-\frac{1}{{z}^{2}}dz$, which is not a section of ${\omega }_{X}$ since it has a pole worse than had f!

Do you have a similar question?