# Projective scheme

A closed sub-scheme of a projective space $\mathbf{P}_{\mathbb{k}}^{n}$. In homogeneous coordinates $x_{0},\ldots,x_{n}$ on $\mathbf{P}_{\mathbb{k}}^{n}$, a projective scheme is given by a system of homogeneous algebraic equations: $${f_{1}}(x_{0},\ldots,x_{n}) = 0, \quad \ldots \quad {f_{r}}(x_{0},\ldots,x_{n}) = 0.$$

Every projective scheme is complete (compact in the case $\mathbb{k} = \mathbb{C}$). Conversely, a complete scheme is projective if there is an ample, invertible sheaf on it. There are also other criteria of projectivity.

A generalization of the concept of a projective scheme is a projective morphism. A morphism $f: X \to Y$ of schemes is called projective (and $X$ is called a projective scheme over $Y$) if and only if $X$ is a closed sub-scheme of the projective fiber bundle ${\mathbf{P}_{Y}}(\mathcal{E})$, where $\mathcal{E}$ is a locally free $\mathcal{O}_{Y}$-module. A composition of projective morphisms is projective. The projectivity of a morphism is also preserved under a base change; in particular, the fibers of a projective morphism are projective schemes (but not conversely). If a scheme $X$ is projective and $X \to Z$ is a finite surjective morphism, then $Z$ is also projective.

Any projective scheme (over $Y$) can be obtained using the construction of the projective spectrum. Restricting to the case of an affine base, $Y = \operatorname{Spec}(R)$, suppose that $\displaystyle A = \bigoplus_{i \geq 0} A_{i}$ is a graded $R$-algebra with the $R$-module $A_{1}$ being of finite type and generating the algebra $A$, and suppose that $\operatorname{Proj}(A)$ is the set of homogeneous prime ideals $\mathfrak{p} \subseteq A$ not containing $A_{1}$. Equipped with the natural topology and a structure sheaf, the set $\operatorname{Proj}(A)$ is a projective $Y$-scheme; moreover, any projective $Y$-scheme has such a form.

Given a vector bundle $E$ over $Y$ (or, equivalently, a locally free $\mathcal{O}_{Y}$-module $\mathcal{E}$), the associated projective bundle, or projective fiber bundle, has as fiber over $y \in Y$ the projective space $\mathbf{P}(E_{Y})$ of all $1$-dimensional sub-spaces of the vector space $E_{Y}$.