In Rick Miranda's book "Algebraic Curves and Riemann Surfaces", in order to prove Plücker's formula for smooth projective plane curves, he first defines a projective plane curve by the formula $X:F(x,y,z)=0$, where of course $F$ is a homogeneous polynomial. Now, he defines the map $\pi:X\to\mathbb{P}^2$ such that $[x:y:z]\mapsto[x:z]$, and uses properties of this map to prove the formula. This may be a really dumb question with an obvious answer, but is this function $\pi$ really defined on all the curve $X$? What if $F(x,y,z)=x+z$; then $\pi[0:1:0]$ wouldn't be defined.
The proof in the book uses the ramification divisor of $\pi$, and so necessarily makes use of the fact that $\pi$ is well defined.
If anyone can clarify this problem for me, I would greatly appreciate it.
Thanks!