Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

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 $f$ !

share|improve this answer
    
Rational sections are global sections? –  Theodore Nov 18 '12 at 21:24
1  
Dear Theodore: no, they are not. I've added an edit in order to clarify this. –  Georges Elencwajg Nov 18 '12 at 21:56
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.