Projective variety
Definition
Let $k$ be an algeraically closed field.
Projective space
The projective $n$-space, denoted $\mathbb{P}^n$, is the affine space of dimension $n+1$ with the origin removed an all lines through the origin identified:
\begin{gather}
(\mathbb{A}^{n+1}-{0})/\sim \
(a_0,\dots,a_n)\sim (\lambda a_0,\dots,\lambda a_n),\quad \lambda\in k\setminus{0}
\end{gather}
The equivalence class of a point $P\in\mathbb{P}^n$ is called the set of homogeneous coordinates for $P$.
Algebraic sets
Let $T\subset k[x_0,\dots,x_n]$. Define $$ Z(T)={P\in\mathbb{P}^n\mid f(P)=0\text{ for all }f\in T}. $$
Definition. A subset $Y\subset\mathbb{P}^n$ is called an algebraic set if there exists a set $T$
Properties
- If $V\subset \mathbb{P}^n$ is algebraic and $V\cap U_i$ is irreducible, then $V$ is irreducible.