Let $C$ be an algebraic curve of genus $g \geq 2$ defined over a number field $K$, having good reduction outside of a finite set $S$ of primes in $K$. A *singly branched* cover $C'$ of $C$ is a curve such that there exists a map $\pi : C' \rightarrow C$ and a point $x \in C(K)$ such that $\pi$ is only branched at $x$. 


Is there a way to relate the Faltings height of $C$ and $C'$? In particular, is the height of $C'$ necessarily dependent on $x$?



Comparing the height of a curve and a singly branched cover
If $f:C'\to C$ is a singly branched cover, the (Faltings) heights of $C$ and $C'$ are certainly related. 


First, by Lemma 6.1 in [1], the inequality $$h(C) \leq h(C') + \log(2\pi) g(C') \log(\deg f)$$ holds. Since $g(C') \leq 3 g(C) \deg f$, this implies that 
$$ h(C) \leq h(C') + 100 g(C) \log(\deg f) \deg f.$$


Presumably, you are more interested in an inequality the other way around.
Using the main result of [2], one can obtain an explicit upper bound for $h(C')$ in terms of the Belyi degree of $C$, the degree of $f$, and the height of $x$. I can explain this in more detail if you'd like.


[1] Javanpeykar, A. *An effective Arakelov-theoretic version of the hyperbolic isogeny theorem.* Math. Proc. Cambridge Phil. Soc., (2016) Vol. 160, Issue 03, 463-476.


[2] Javanpeykar, A. *Polynomial bounds for Arakelov invariants of Belyi curves, with an appendix by Peter Bruin.*
Algebra and Number Theory, Vol. 8 (2014), No. 1, 89-140.


