Plane curve

Remarks

The polynomial is determined by $C$ up to a nonzero scalar factor. A plane curve is an algebraic curve: it has dimension $1$. It is reduced exactly when $F$ has no repeated irreducible factor, and integral exactly when $F$ is irreducible. Example: $F = x^2 + y^2 - z^2$ defines a conic of degree $2$ in $\mathbb{P}^2$, the projective closure of the circle $x^2 + y^2 = 1$. Non-example: the twisted cubic $\set{[s^3 : s^2 t : s t^2 : t^3]} \subset \mathbb{P}^3$ is a curve in $\mathbb{P}^3$ that lies in no plane of $\mathbb{P}^3$, so it is not a plane curve inside $\mathbb{P}^3$. As an abstract curve it is isomorphic to $\mathbb{P}^1$, which is a line in $\mathbb{P}^2$; being a plane curve is a property of the embedding. A nonzero linear form vanishing on it would give a linear relation among $s^3, s^2 t, s t^2, t^3$, and there is none.