Rational maps from $\mathbb{P}^{1}$ to $\mathbb{P}^{1}$

Question

I was reading Arithmetic of Elliptic Curves by Silverman,I had the following question:
Is there a way to classify rational maps from $\mathbb{P}^{1}$ to $\mathbb{P}^{1}$ over some algebraically closed field K?

Now,rational maps between non-singular varieties of dimension one are morphisms.Then $\phi:\mathbb{P}^{1}\longrightarrow\mathbb{P}^{1}$ is a morphism with deg$\phi$=1.Then $\phi$ has to be an isomorphism.Hence,rational maps between $\mathbb{P}^{1}$ and $\mathbb{P}^{1}$ must be isomorphisms.

Is this a correct reasoning or am I wrong somewhere?

Answer 1

It is not true that $\phi$ must be an isomorphism, nor that $\deg\phi$ must be 1. For instance, $[x:y]\to [x^2:y^2]$ is a map which is of degree 2 and is not an isomorphism.

In general, a map $\Bbb P^1\to \Bbb P^1$ is given by a two coprime homogeneous polynomials of the same degree.