---
tag: J7NBAJMQND
kind: definition
name: Projective variety
msc: [14N05, 14A10]
defines: A4H4CZ2V35
uses: [0PF6TM8TWT, ZT9A6JK8QF, 2C3PAQ7H50, 6ZBNGS0P98, SY2MWC434Z, VHFMQQYSTN, JERVQT4C12]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_variety
---

## Definition

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]], let $n \ge 0$, and let $\mathbb{P}^n$ be the [[projective space|ZT9A6JK8QF]] over $k$.
For a set $T$ of [[homogeneous polynomials|2C3PAQ7H50]] in $k[x_0, \dots, x_n]$, let
$$V(T) = \set{[a] \in \mathbb{P}^n : f(a) = 0 \text{ for all } f \in T}.$$
This is well defined, because $f(\lambda a) = \lambda^d f(a)$ when $f$ is homogeneous of degree $d$.
A *projective algebraic set* is a set of the form $V(T)$; these are the closed sets of the Zariski topology on $\mathbb{P}^n$.
A *projective variety* is a projective algebraic set that is [[irreducible|SY2MWC434Z]] in this topology.

## Remarks

The ideal of a projective algebraic set $X$ is the [[homogeneous ideal|6ZBNGS0P98]] generated by the homogeneous polynomials that vanish on $X$.
Then $X$ is irreducible if and only if this ideal is prime.

Example: the plane conic $C = V(xz - y^2) \subseteq \mathbb{P}^2$ is a projective variety.
Its polynomial $xz - y^2$ is irreducible, so it generates a prime ideal, and $C$ is the image of $\mathbb{P}^1$ under $[s : t] \mapsto [s^2 : st : t^2]$.

Non-example: $V(xy) \subseteq \mathbb{P}^2$ is a projective algebraic set but not a projective variety.
It is the union of the two lines $V(x)$ and $V(y)$.

The intersection of $X$ with a chart $U_i = \set{x_i \ne 0} \cong k^n$ is an [[algebraic set|VHFMQQYSTN]] in $k^n$, defined by setting $x_i = 1$ in the equations; when nonempty, it is an [[affine variety|JERVQT4C12]].
Some authors do not require irreducibility.