I'm self-studying the module, Belyi Maps and Dessins d'Enfants. At the end of lecture 1's notes link, there is a motivating example for the subsequent study.
But, at the second line, I am finding it difficult to link the derivative $F_y (P) \neq 0$ and the projection $(x, y) \mapsto x$ onto the x-axis becoming a chart. Is there any result connecting them or am I missing something else? Could anyone help me to understand?
** This relation is actually used subsequently to prove the Lemma 4 of Lecture 4's notes;
The implicit function theorem states that there exists a function $g(x)$ such that $0 = f(x,y) \Longleftrightarrow y=g(x)$ in a neighbourhood of the point $(x,y)$.
In other words, the projection $\pi_x:U\to V:(x,y)\mapsto x$ is bijective (since the points $(x,y)$, with $f(x,y)=0$, are exactly the points $(x,g(x))$ and thus completely determined by its $x$-value), meaning that $\pi_x$ maps the points of that neighbourhood one-to-one to a subset of $\mathbb{R}$, wich makes it a (1-dimensional) chart.