Is the cuspidal curve $\mathcal{M}$ is a coarse moduli space for lines in $\mathbb{C}^2$?

Question

As the question suggests, is the cuspidal curve $\mathcal{M}$ a coarse moduli space for lines in $\mathbb{C}^2$? I'm inclined to believe the answer is no, but all attempts at proving it so far have seemed not fruitful...

Answer 1

So, I assume you mean "lines through the origin" because otherwise, a dimension count suffices (lines are $ax+by+c=0$ and so have three parameters, but modulo a $\mathbb{C}^\times$ gives a $2$ dimensional family). A quick, but admittedly non-rigorous (though it can be made rigorous) way to see that it is not is that the moduli space of lines through the origin in $\mathbb{C}^2$ must be smooth, because there are automorphisms of $\mathbb{C}^2$ that swap any two lines through the origin, thus the automorphism group of the moduli space of them acts transitively, and so it is smooth.