Projective conic

Remarks

Example: $Q(x, y, z) = xz - y^2$. Its points are $[1 : t : t^2]$ for $t \in F$ and the point $[0 : 0 : 1]$: if $x \ne 0$ scale to $x = 1$ and get $z = y^2$; if $x = 0$ then $y = 0$. It is non-degenerate. If $Q$ involved at most two variables in some coordinates, there would be a nonzero vector $v$ with $Q(u + v) = Q(u)$ for all $u$. Here $Q(u + v) - Q(u) = u_x v_z + u_z v_x - 2 u_y v_y + Q(v)$, and this vanishes for all $u$ only if $v_x = v_z = 0$ and $Q(v) = -v_y^2 = 0$, so $v = 0$. If $F = \F_q$, this conic has $q + 1$ points.

Non-example of non-degeneracy: $Q(x, y, z) = xy$ gives the union of the two lines $x = 0$ and $y = 0$. It involves only $x$ and $y$, so it is degenerate.

Over $\R$, the non-degenerate form $x^2 + y^2 + z^2$ has an empty conic, so the form carries more information than the point set. Over an algebraically closed field, or over $\R$ when the conic has a point, all non-degenerate conics are equivalent under projective transformations. The affine conic sections are the parts of projective conics off the line at infinity.