Algebraic variety (awaiting review)

There are two main kinds of algebraic variety, affine varieties and projective varieties.

A (reduced) affine variety defined over the field $K$ is a subset $V$ of some affine space $\mathbb A^n(\overline K)$ such that $$ I(V)= \{f\in \overline{K}[X_1, \ldots, X_n] : f(P) = 0 \text{ for all } P \in V \}$$ is a prime ideal in $\overline{K}[X_1, \ldots, X_n]$ that can be generated by polynomials in $K[X_1, \ldots, X_n].$

A (reduced) projective variety defined over the field $K$ is a subset of some projective space $\mathbb P^n(\overline K)$ such that $$I(V) = \{f\in \overline K[X_0,X_1, \ldots, X_n] : f \text{ homogeneous and }f(P) = 0 \text{ for all } P \in V \}$$ is a homogeneous prime ideal in $\overline K[X_0, X_1, \ldots, X_n]$ (that is, an ideal that can be generated by homogeneous polynomials).