Projective variety

Remarks

The ideal of a projective algebraic set $X$ is the homogeneous ideal 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 in $k^n$, defined by setting $x_i = 1$ in the equations; when nonempty, it is an affine variety. Some authors do not require irreducibility.