I am reading Silverman's The Arithmetic of Elliptic Curves. I have a question regarding Example 3.5, which is the following:
I have doubts about the equality $[-Y^2,Y(X-Z))]=[-Y,X-Z]$. Two points $[x_0,\ldots, x_n]$,$[x_0',\ldots, x_n']$ in the projective space are the same if we can write $x_i=\lambda x_i'$ for some $\lambda \in K^\times$ for every $i=0\ldots, n$. So my question here is: what happens if $Y=0$? I don't think we can safely state that $[-Y^2,Y(X-Z))]=[-Y,X-Z]$. Actually, if $Y=0$, we have that $[-Y^2,Y(X-Z))]=[0,0]$ so this is not even a well-defined point. What worries me is that this reasoning he does is precisely meant to solve a problem that arises precisely when $Y=0$, which is the only time when I don't see his reasoning working.
Be careful: the point is not that the equality $[-Y^2,Y(X-Z)]=[-Y,X-Z]$ holds at a point with $Y=0$ - rather, this equality holds on their common domain of definition and you can consider the map which is the gluing of these maps.
Said slightly differently, you can consider a map defined by $[X+Z,Y]$ on the open set $V\setminus [1,0,-1]$ and $[-Y,X-Z]$ on the open set $V\setminus [1,0,1]$, and since $[X+Z,Y]=[-Y,X-Z]$ on the open set $V\setminus \{[1,0,-1],[1,0,1]\}$, these formulas combine to give one map $V\to\Bbb P^1$.