While working on a project for mathematics I came across the following lemma: [Kock]


If $X$ is a curve defined over an algebraically closed subfield $N$ of $\mathbb{C}$ and let $t:X\rightarrow \mathbb{P}\_{\mathbb{C}}^1$ be a finite morphism defined over $N$, then the critical values of $t$ are $N$-rational. 


I have only recently started learning about this sort of stuff and have been utlising Forster's *Lectures on Riemann Surfaces*. I restated the theorem as follows:


If $X$ is a compact connected Riemann surface defined over an algebraically closed subfield $N$ of $\mathbb{C}$ and let $t:X\rightarrow \mathbb{P}\_{\mathbb{C}}^1$ be a non-constant holomorphic map defined over $N$, then the critical values of $t$ are $N$-rational. 


I believe this is allowable as compact Riemann surfaces can be embedded in a projective space where they are the common zero of a set of polynomials over $N$. (My wording here may be a little off - feel free to correct me).


Kock in his proof uses the following fact that seems simple but has me absolutely stumped (even after days of research):


The set of critical values of $t$ are given by $t(supp(\Omega\_{X/\mathbb{P}\_{\mathbb{C}}^1}))$


What is the definition of $supp(\Omega\_{X/\mathbb{P}\_{\mathbb{C}}^1})$? And how does this fact follow?



Branch points of a non-constant holomorphic map between compact riemann surfaces
This is quite standard and probably the question belongs to <http://math.stackexchange.com> rather than Mathoverflow, anyway let me give an answer.


The holomorphic map $t \colon X \to \mathbb{P}^1$ induces a natural map $dt$ between tangent bundles, hence a short exact sequence of coherent sheaves on $X$: $$0 \longrightarrow T\_X \stackrel{dt}{\longrightarrow} t^\*T\_{P^1} \longrightarrow T\_{X/P^1} \longrightarrow 0,$$
where the cokernel $T\_{X/P^1}$ is called the *relative tangent sheaf* of $t$.


Notice that, by the Jacobian criterion, the critical points of $t$ are precisely the points were $dt$ has not maximal rank, so the critical values (branch points) of $t$ are given by
$t(\textrm{Supp}(T\_{X/P^1}))$.


Dualizing the previous sequence one obtains


$$0 \longrightarrow t^\*\Omega\_{P^1} \stackrel{(dt)^\*}{\longrightarrow} \Omega\_X \longrightarrow \Omega\_{X/P^1} \longrightarrow 0,$$


where the map $(dt)^\*$ is induced by the pullback of the holomorphic $1$-forms and the cokernel $\Omega\_{X/P^1}= \operatorname{Ext}^1(T\_{X/P^1}, \, \mathcal{O}\_X)$ is called the *relative cotangent sheaf*.


Since clearly $\textrm{Supp}(T\_{X/P^1})=\textrm{Supp}(\Omega\_{X/P^1})$, the claim follows.


