Consider a cubic plane curve like this:
$$y(y - x^2 - 1) = 0$$
which is clearly the "union" of a line ($y = 0$) and a parabola ($y = x^2 + 1$) that have no points in common (not even the point at infinity), what kind of curve is this? How do I classify it?
This is a degree 3 curve, and I'm unable to find any singularity, so it should be of genus 1. However, I'm not fully convinced by that, and I think I'm making a mistake: intuitively, this should allow a rational parametrization, so it should be of genus 0. Can somebody please shed some light on this?