What is the nature of this surface?

Question

What is the nature of the surface whose equation is (it depends on $m$) $$x^2+2y^2+(m+1)z^2+2xy-2yz-2x+2y-4z+m^2+4=0$$

Answer 1

The symmetric matrix corresponding to your quadratic form is $$A = \begin{pmatrix} 1 & 1 & 0 & -1 \\ 1 & 2 & -1 & 1 \\ 0 & -1 & m+1 & -2 \\ -1 & 1 & -2 & m^2 + 4 \end{pmatrix}.$$

A straightforward calculation reveals that $\det(A) = m^3 - m$. Thus, for $m \neq -1, 0, 1$, the surface is a smooth quadric surface (even when the surface is completed as a quadric in $\mathbb{P}^3$). When $m = -1, 0, 1$, $\text{rank}(A) = 3$ in all cases, indicating that the (completed) surface has a singular point. When $m = \pm 1$, the singular point is $(3,-2,0)$. When $m = 0$, the singular point is on the plane at infinity (so the affine patch of the surface you are looking at it is, in fact, smooth).