The motivation behind this question is the study of hyperbolic PDE (in particular Leray hyperbolicity).
One definition for a polynomial $P(\xi^0, \ldots, \xi^n)$ of degree $m$ to be hyperbolic is that there exist a point $\xi \in \mathbb{R}^{1 + n}$ such that every line through $\xi$ which does not intersect the origin intersects the zero level set $\{P = 0\}$ in exactly $m$ distinct points.
Question: Does this require that one considers the projectivization of $\mathbb{R}^{1 + n}$?
If it does not, then for instance the symbol $\xi_0^2 - \xi_1^2$ of the wave equation in Minkowski spacetime is not hyperbolic, since a line parallel to the light cone's boundary would only intersect the cone once. But I have not seen this explicitly stated in the literature.