Definition:Projective Algebraic Set
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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
Sources
- 1977: Robin Hartshorne: Algebraic Geometry $\text I.2$ Projective Varieties