TITLE: Holomorphic mappings, Plücker's formula (Mistake in Rick Miranda's book?)
QUESTION [8 upvotes]: 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!

REPLY [9 votes]: The proof starts with the words " Let $X$ be a smooth projective curve...". And smooth is the magic word that solves your problem. Indeed there is a proposition  stating that a rational (or meromorphic) map from a smooth curve to projective space is actually regular (or holomorphic). 
In your example you have, for a point $[x:y:z]\in X$  different from $P=[0:1:0]$, the formula $\pi [x:y:z]=[x:z]=[x:-x]=[1:-1]$,  taking into account that $x+z=0$  on $X$. So it is easy to extend $\pi$ regularly through $P$, since $\pi$ is actually constant. You just define $\pi(P)=\pi( [0:1:0])=[1:-1]$ .
Remark The proposition mentioned above (on regularity of rational maps)  is not difficult. If you take a coordinate $t$ for your curve near the troublesome point $P$, locally at $P \;$ the map is $t\mapsto [t^au(t):t^bv(t)]$ (where I have assumed that the target is $\mathbb P^1$) with $a,b\in \mathbb N$ and $u,v\in\mathcal O_{X,P}^\ast$  (units of $\mathcal O_{X,P})$ . If $a>b$  the map is just $t\mapsto [t^au(t):t^bv(t)]=[t^{a-b}u(t):v(t)]$ and can clearly be extended regularly by sending $P$ to $[0:v(0)]=[0:1]$ . And similarly if $a\leq b$ . And similarly if the target is $\mathbb P^N$.