# Classifying complex conics up to isomorphism as quotient rings of $\mathbb{C}$

This is a continuation of the question I asked here. The problem is now:

Let $Q = ax^2 + bxy + cy^2 + dx + ey + f \in \mathbb{C}[x,y]$ be a general quadratic polynomial, that is, $a,b,c \not= 0$. A change of coordinates allows this to be rewritten as $Q = x^2 + cy^2 + ey + f$. Classify the rings $\mathbb{C}[x,y]/(Q)$ up to isomorphism. For each isomorphism class, draw a picture of a corresponding variety. Determine if it is reducible and/or connected, and whether $(Q)$ is radical.

I’ve tried to factor the equation $Q = 0$ in the following way:

$$Q = 0 \implies x^2 + c \left( y + \frac{e}{2c} \right)^2 = \frac{e^2}{4c} -f$$
Now, if either $e$ or $f$ is nonzero then this is the equation for an ellipse. It should be straightforward to find a parametrizaton using sine and cosine, that is, we should get $\mathbb{C}[x,y]/(Q) \simeq \mathbb{C}[\cos{t},\sin{t}]$.

On the other hand, if $e$ and $f$ are both zero, then the equation
$$x^2 + cy^2 = (x + i\sqrt{c}y)(x – i\sqrt{c}y)=0$$
can be factored, so this is a reducible polynomial. In this case, $y$ is just $x$ times a constant, so $\mathbb{C}[x,y]/(Q)$ should be isomorphic to $\mathbb{C}[x]$.

And… that’s it! Those are the only two cases I could find. But I feel like I must have missed something very important. In $\mathbb{R}$, there are many kinds of conics: hyperbolas, ellipses, parabolas, and degenerate cases. Surely there can’t just be two cases in $\mathbb{C}$?

I’m also struggling with the last part of the problem: I have no idea how to draw a complex curve (four dimensions required?) and I don’t know how to determine if a variety is connected. I think the first case (the ellipse) is an irreducible polynomial, so the variety is also irreducible and the corresponding ideal is radical. The second case (the reducible polynomial) is a reducible variety, but does that mean the ideal is not radical?

#### Solutions Collecting From Web of "Classifying complex conics up to isomorphism as quotient rings of $\mathbb{C}$"

The classification of conics in $\mathbb A^2$ shows that there are the following possibilities:

1. $\mathbb C[X]$ and $\mathbb C[X,X^{-1}]$ (for the irreducible ones);

2. $\mathbb C[X]/(X^2)$, $\mathbb C[X]/(X^2-1)\simeq\mathbb C\times\mathbb C$, and $\mathbb C[X,Y]/(X^2-Y^2)\simeq\mathbb C[X,Y]/(XY)$ (for the reducible ones).