A birational map from $\mathbb{P}^1$ to an irreducible plane projective curve

Question

Let $C$ be an irreducible plane projective curve described by the equation $$zf(x, y) + g(x, y) = 0,$$ where $f$ and $g$ are a homogenous forms of degree $d - 1$ and $d$, respectively. What would be the birational map $\mathbb{P}^1 \to C$?

Answer 1

As you noticed, one can try the function $$F:(x:y)\in P^1\longmapsto \bigl(x:y:-\tfrac{g(x,y)}{f(x,y)}\bigr)\in P^2.$$ This has image contained in the curve $C$. It is important that this is well-defined: the point $(x:y)$ is equal to $(\lambda x:\lambda y)$, and the points $(x:y:-\tfrac{g(x,y)}{f(x,y)}\bigr)$ and $(\lambda x:\lambda y:-\tfrac{g(\lambda x,\lambda y)}{f(\lambda x,\lambda y)}\bigr)$ are also equal, precisely because $f$ and $g$ are homogeneous of the degrees they have.

Notice that the map is defined only at the points $(x:y)$ of $P^1$ where $f(x,y)\neq0$; this is not a big problem: there are finitely many such points. The domain of our function is the open set which is complementary to the zero set of $f$.

On the other hand, we have a function $$G:(x:y:z)\in C\longmapsto (x:y)\in P^1.$$ You should find exactly where this is defined and check that it is actually well-defined. Finally, you should check that $F$ and $G$ are inverse in the appropriate sense.