# the divisor of a rational function

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?

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Yes, $df$ is a rational section of $\omega_X$, which one could write rigorously as $df\in \Gamma(X,\omega _X\otimes _{\mathcal O_X} \mathcal K_X)$.
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 $\mathbb P^1_\mathbb C$) which is not the differential of any rational function.

Edit
Of course if $f$ is not regular (i.e. if $f$ has poles) $df$ will not be a global section of $\omega_X$ : $df\in \Gamma(X,\omega _X\otimes _{\mathcal O_X} \mathcal K_X)\setminus \Gamma(X,\omega _X)$
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 $f$ !

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Rational sections are global sections? – Theodore Nov 18 '12 at 21:24
Dear Theodore: no, they are not. I've added an edit in order to clarify this. – Georges Elencwajg Nov 18 '12 at 21:56