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

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- Review status: beta
- Last edited by David Farmer on 2018-08-20 15:03:29

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