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 $K[x_1,\dots,x_n]$.

**Knowl status:**

- Review status: reviewed
- Last edited by John Voight on 2020-10-11 12:44:07

**Referred to by:**

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

- 2020-10-11 12:44:07 by John Voight (Reviewed)
- 2020-10-11 12:02:30 by John Voight
- 2020-10-11 12:02:00 by John Voight
- 2020-10-11 12:01:50 by John Voight
- 2020-10-11 11:35:07 by John Voight
- 2020-10-11 11:22:18 by John Cremona
- 2020-10-11 11:21:55 by John Cremona
- 2020-10-11 10:57:30 by John Voight
- 2020-10-10 11:05:27 by Andrew Sutherland
- 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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