Definition:Projective Algebraic Set

Definition

Let $K$ be a field.

Let $n \in \N_{>0}$.

Let $\map {\mathbb P^n} K$ be the $n$-projective space over $K$.

Let $S = K \sqbrk {X_0, \ldots, X_n}$ be the ring of polynomial functions in $n + 1$ variables over $K$.

Then a subset $X \subseteq \map {\mathbb P^n} K$ is a projective algebraic set if and only if:

there is a subset $T \subseteq S$ of homogeneous elements such that:

$\ds X = \bigcap_{f \mathop \in T} \set {\struct {x_0 : \cdots : x_n} \in \map {\mathbb P^n} K : \map f {x_0, \ldots, x_n} = 0}$

Also see

• Definition:Affine Algebraic Set

Sources

• 1977: Robin Hartshorne: Algebraic Geometry $\text I.2$ Projective Varieties