What are the different types of degenerate conics and other algebraic curves?

Question

What are the different types of degenerate conics and other algebraic curves?

Consider $$Ax^2 + 2Bxy + Cy^2 + Dx + Ey + F = 0.$$

This can fail to be a standard conic in multiple ways:

1. $A = B = C = 0$. In this case, the degree is 1, and the curve isn't a conic at all. This is not considered degenerate.

2. $A > 0, B = 0, C > 0, F < 0$. This is an imaginary ellipse, also not considered degenerate.

3. $A > 0, B = 0, C > 0, F = 0$. This is degenerate, because it can be factored (?).

Is that correct? "Degenerate" proper is when the polynomial can be factored. If it's not degenerate, and is of the full degree, it can still not correspond to the standard geometric curve if (but only if) it's imaginary?

Answer 1

Since you are assigning signs to the coefficients, you seem to be dealing with real algebraic curves. In such case, the conic $x^2+y^2+1=0$, i.e., the empty set, will certainly be considered degenerate, and the rule of thumb is that a real conic is nondegenerate iff it is an ellipse, parabola, or hyperbola, or equivalently if it can be represented by an intersection in $\mathbb R^3$ of a standard cone and a plane not passing through the origin. Of course if you want to complexify the situation, or consider the projectivized curve, the situation will be different as mentioned in the other answer.

Answer 2

The best way to describe a degenerate conic is that it's a conic which factors as a product of two lines over an algebraically closed field. Your classification does not include any reference to that, and it causes you to miss things like $x^2+y^2$ which gives you a single point over $\Bbb R$ which really ought to be considered degenerate.

Your list of conditions which correspond to degeneracies is not complete (for instance, there are degenerate conics which do not satisfy $A>0,B=0,C>0,F=0$ like $x^2+xy=x(x+y)$; there are imaginary ellipses with nonzero $B$; etc.). The best way to check whether a conic is degenerate or not is to compute a determinant, as described on the wikipedia page: the conic $Ax^2+2Bxy+Cy^2+2Dx+2Ey+F=0$ is degenerate iff $$\det \begin{pmatrix} A & B & D \\ B & C & E \\ D & E & F \end{pmatrix} = 0.$$ This is because this matrix condition corresponds to a singular point on the projective closure of your conic over an algebraically closed field, and this happens iff your conic decomposes as a union of two lines (this may require you to learn a bit more algebraic geometry to fully understand this).