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.

Comments

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} $.

References

[1]	D. Mumford, “Algebraic geometry”, 1. Complex projective varieties, Springer (1976). MR0453732 Zbl 0356.14002
[a1]	R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 91. MR0463157 Zbl 0367.14001