Quadric

Remarks

Example: in $\mathbb{P}^3(F)$, the form $Q(x) = x_0 x_1 + x_2 x_3$ defines a quadric that contains the line of points $[s : 0 : t : 0]$, since $Q(s, 0, t, 0) = 0$ for all $s, t$. Example: the sphere $x^2 + y^2 + z^2 = 1$ in $\R^3$ is an affine quadric, and $x^2 + y^2 + z^2 - w^2 = 0$ is its projective closure.

Non-example: the zero set of $y - x^3$ in $F^2$ is not an affine quadric, because the polynomial has degree $3$.

For $n = 2$ the projective quadrics are the projective conics. Different forms can define the same point set. For example, over $\R$ every positive definite form defines the empty quadric. For this reason the quadric is often taken together with its form, up to a nonzero scalar.