Characterize the values of A and B for which variety is singular Let $A, B ∈ \overline{K}$ with $K$ a field. Characterize the values of A and B for which each of the following variety is singular:
$V : Y^2 Z + AXY Z + BY Z^2 = X^3$
Looking at the point where partial derivatives vanish, we have (0,0,0) is always a singular point, which means that for any pair $(A,B)\in \overline{K}^2$, V is singular. Is that correct?
Thank you for any help or hint.

Well, I'm sure you're talking about a projective curve (single homogeneous equation in $X,Y,Z$) whose points lie in the projective plane. Note that the point $(0,0,0)$ does not belong to the projective plane.
