Projective space
Algebraic sets
Recall that a polynomial $F\in k[x_0,\dots,x_n]$ is called homogeneous of degree $d$ if $F(\lambda x_0,\dots,\lambda x_n)=\lambda^d\cdot F(x_0,\dots,x_n)$.
If $T$ is a set of homogeneous elements of $S$, define the zero set of $T$: $$Z(T)={P\in\mathbb{P}^n\mid f(P)=0\text{ for all }f\in T}.$$
Let $S$ denote the polynomial ring $k[x_0,\dots,x_n]$ regarded as a graded ring. (See Polynomial ring for details.)
Definition. A subset $Y$ of $\mathbb{P}^n$ is an algebraic set if there exists a set $T$ of homogeneous elements of $S$ such that $Y=Z(T)$.
One can show that these are closed under finite unions and arbitary intersections, allowing us to define the Zariski topology.
Definition. A quasi-projective algebraic set is an open subset of a (projective) algebraic set.
Examples
- On $\mathbb{P}^1$, the projective algebraic sets are $\emptyset$, $\mathbb{P}^1$, and finite collections of homogeneous points ${p_1,\dots,p_n}$.