I am currently going through the appendix A on Projective Geometry of Silverman & Tate Rational Points on Elliptic Curves. In the section about intersection of curves on the projective plane, the authors give the following example:
Find the intersection points of the following two curves (a parabola and a line):
$$C_1 : x + 1 = 0 \text{ and } C_2: x^2 - y = 0$$
The first intersection point is simple to see, it is $(-1,1)$. We now need to find to point of intersection at infinity. We first homogenize the two equations to obtain (reusing notation):
$$C_1 : X + Z = 0 \text{ and } C_2: X^2 - YZ = 0$$
This is where my understanding fails to follow what the book says. From what I get, we want to look at the points at infinity by setting Z = 0. The book however says the following:
Then $C_1\cap C_2$ consists of the two points $[−1, 1, 1]$ and $[0, 1, 0]$, as may be seen by substituting $X = −Z$ into the equation for $C_2$.
For me, this does not make sense for if we substitute as it is instructed, we get that Y = 0 which cannot happen in homogeneous coordinates. Can someone help me resolve this gap in my understanding and find the $[0,1,0]$ point given by the authors? I have tried to read multiple ressources here and they do seem to agree with my method of finding the points at infinity by setting $Z=0$.