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

Beginner2022-02-16Added 6 answers

Yes, df is a rational section of $\omega}_{X$ , which one could write rigorously as $df\u03f5\mathrm{\Gamma}(X,{\omega}_{X}{\otimes}_{{O}_{X}}{K}_{X})$ .

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 $P{P}_{\left\{\mathbb{C}\right\}}^{1}$ ) which is not the differential of any rational function.

Beware however that not all rational sections of

an2gi2m9gg

Beginner2022-02-17Added 9 answers

Edit:

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

$df\u03f5\mathrm{\Gamma}(X,{\omega}_{X}{\otimes}_{{O}_{X}}{K}_{X})\text{}\mathrm{\Gamma}(X,{\omega}_{X})$

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!

Of course if f is not regular (i.e. if f has poles) df will not be a global section of

For example if