The Stacks project

Exercise 111.39.2. Let $k$ be a field. Let $Z \subset \mathbf{P}^2_ k$ be an irreducible and reduced closed subscheme. Show that either (a) $Z$ is a closed point, or (b) there exists an homogeneous irreducible $F \in k[X_0, X_1, X_2]$ of degree $> 0$ such that $Z = V_{+}(F)$, or (c) $Z = \mathbf{P}^2_ k$. (Hint: Look on a standard affine open.)