So I'm trying to understand a proof of Belyi's theorem from <http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf>


specifically lemma 3.4.


The setup is as follows: Let $X/\mathbb{C}$ be a curve, and let $t : X\rightarrow\mathbb{P}^1\_\mathbb{C}$ be a meromorphic function on $X$ thought of as a covering map of degree $n$. Further, let $\text{Crit}(t)$ denote the critical points of the cover $t$ - ie, the points in $\mathbb{P}^1\_\mathbb{C}$ which have fewer than $n$ pre-images under $t$.


Then, he claims that $\text{Crit}(t) = t(\text{supp}(\Omega^1\_{X/\mathbb{P}^1\_\mathbb{C}}))$.


Now, it's my understanding that the critical points of $t$ should be the images of the ramification points of $t$ under $t$, so I've been trying to understand why it should be the case that the sheaf of relative differentials of $X/\mathbb{P}^1\_\mathbb{C}$ should be nonzero only on the ramification points (or at least only above the critical points).


To this end, I'm trying to understand the definition given in Hartshorne (III.8), namely:
$\Omega^1\_{X/\mathbb{P}^1\_\mathbb{C}} = \Delta^\*(\mathcal{I}/\mathcal{I}^2)$, where $\Delta : X\rightarrow X\times\_{\mathbb{P}^1\_\mathbb{C}} X$ is the diagonal map, and $\mathcal{I}$ is the sheaf of ideals of the image $\Delta(X)$ in some open subset $W\subset X\times\_{\mathbb{P}^1\_\mathbb{C}} X$.


I kind of understand sheaves of ideals (they're essentially functions on the ambient space that vanish on the closed subscheme), but I'm still not very comfortable with the notion of $\Delta^\*(\mathcal{I}/\mathcal{I}^2)$ (in this case defined to be $\Delta^{-1}\mathcal{I}/\mathcal{I}^2\otimes\_{\Delta^{-1}\mathcal{O}\_{X\times X}}\mathcal{O}\_X$, where the fibred product is taken over $\mathbb{P}^1$).


Any comments on how I should think of $\Delta^\*(...)$ and why the sheaf of relative differentials only have nonzero stalks at ramification points would be awesome!


thanks.



trying to understand the support of the sheaf of relative differentials
Ravi Vakil has a good explaination for the definition $\Delta^\*(I/I^2)$ in his notes. See his AG notes [here](http://math.stanford.edu/~vakil/0506-216/216class38.pdf) or [here](http://math.stanford.edu/~vakil/216blog/FOAGapr1312public.pdf) (chapter 23). In particular, I guess thinking about this locally makes it a little clearer what's going on, in terms of derivations etc. Also, when $X$ is smooth, it is instructive to see that this really gives the cotangent bundle on $X$. 


As for your question about ramification points: Let $f:X\to Y$ be a finite morphism of curves (I will assume that these are smooth in the following). It is useful to have in mind the exact sequence $$0\to f^\*\Omega\_{Y}\to \Omega\_X \to \Omega\_{X|Y}\to 0.$$(This is exact at the right in the smooth case, but not in general). Note that $\Omega\_{X|Y}$ is a torsion sheaf since the two other sheaves are locally free of the same rank (they are line bundles on $X$). At a point $q\in Y$ and $p\in X$ in the preimage of $q$, let $dx$ denote a generator for $\Omega\_{Y,q}$ as a $O\_{Y,q}$-module. Now, $(\Omega\_{X|Y})\_P=0$ if and only if $f^\*dx$ is a generator of $\Omega\_{X,p}$, which happens if and only if $f$ pulls back a local parameter to a local parameter, that is $p$ is unramified. Moreover, the exact sequence above shows that the ramification index is exactly the length of the sheaf $\Omega\_{X|Y}$. Finally, note that this sequence gives the Riemann hurwitz formula, relating the canonical divisors of $X$ and $Y$ and the ramification divisor of $f$.


