Projective morphism

Remarks

For a field $k$ and a homogeneous polynomial $F \in k[x_0, x_1, x_2]$ of positive degree, the curve $X = V(F) \subseteq \mathbb{P}^2_k$ is projective over $\operatorname{Spec} k$. Indeed, $X \to \mathbb{P}^2_k$ is a closed immersion and $\mathbb{P}^2_k \to \operatorname{Spec} k$ is the projection $p$. The affine line $\mathbb{A}^1_k \to \operatorname{Spec} k$ is not projective. A projective morphism is proper, and $\mathbb{A}^1_k$ is not proper over $k$. This is the definition in Hartshorne's textbook. In EGA the definition is more general: $X$ is a closed subscheme of the projective bundle of a quasi-coherent $\mathcal{O}_Y$-module of finite type. The two definitions agree when $Y$ is affine.