Given the parabola $y-x^2=0$ in $\mathbb{C}^2$, I'm trying to show that in $\mathbb{P}^2$, any line intersects the parabola twice (up to multiplicity).
This is simple enough for the "finite" components of projective space, but I'm having trouble showing this for the point at infinity.
You need projective corrdinates $[x,y,z]$ then the parabola is $$yz=x^2$$ and if the line is represented as $$[\lambda a_1+\mu b_1,\lambda a_2+\mu b_2,\lambda a_3+\mu b_3]$$ Then you have
$$(\lambda a_3+\mu b_3)(\lambda a_2+\mu b_2)=(\lambda a_1+\mu b_1)^2$$ If $a$ and $b$ are distinct points the not all coefficients are zero. Thus this is a quadratic equation in $\lambda$ and $\mu$ and thus has two solutions, with multiplicity ie. it factors.
The line at infinity is given by $z=0$ so the intersection is $$x^2=0$$ this it parabola intersects it once with multiplicity $2$.