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).

**Authors:**

**Knowl status:**

- Review status: beta
- Last edited by John Cremona on 2019-10-30 11:46:27

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**History:**(expand/hide all)

- 2019-10-30 11:46:27 by John Cremona
- 2019-09-04 17:47:19 by John Jones
- 2018-08-20 15:03:29 by David Farmer

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