Algebraic variety (reviewed)

There are two main kinds of algebraic varieties, affine varieties and projective varieties. Both are defined as the set of common zeros of a collection of polynomials. Let $K$ be a field with algebraic closure $\overline{K}$.

An affine algebraic set is a subset of affine space $\mathbb A^n(\overline K)$ of the form $$ V(I) = \{P \in \mathbb A^n(\overline K) : f(P) = 0\text{ for all }f \in I\} $$ where $I \subseteq \overline{K}[x_1,\dots,x_n]$ is an ideal. Given an affine algebraic set $V$, its defining ideal is $$ I(V) = \{ f \in \overline{K}[x_1,\dots,x_n] : f(P)=0\text{ for all }P \in V\}.$$

An affine variety over $\overline{K}$ is an affine algebraic set whose defining ideal $I \subseteq \overline{K}[x_1,\dots,x_n]$ is a prime ideal. An affine variety over $K$ is an affine variety over $\overline{K}$ whose defining ideal can be generated by polynomials in $K[x_1,\dots,x_n]$.

We define projective notions similarly. A projective algebraic set is a subset of projective space $\mathbb P^n(\overline{K})$ defined by a homogeneous ideal $I \subseteq \overline{K}[x_1,\dots,x_n]$. A projective variety over $\overline{K}$ is a projective algebraic set whose defining ideal is a homogeneous prime ideal. A projective variety over $K$ is a projective variety over $\overline{K}$ whose defining ideal can be generated by homogeneous polynomials in $K[x_1,\dots,x_n]$.