The complex projective space $\mathbb{P}^n$ is also a projective variety. According to Hartshorne, a projective variety is defined a s the zero set of a subset of homogeneous polynomials defined on $\mathbb{C}^n$ (he uses a generic algebraically close field $k$ but I prefer to study $\mathbb{C}$.
Can you give me an example of such polynomials whose zero set is understood as $\mathbb{P}^n$?
That is, I would especially like to understand the cases of $\mathbb{P}^2$, $\mathbb{P}^1 \times \mathbb{P}^1$ (and more general Hirzebruch surfaces) as zero sets of polynomials over $\mathbb{C}[x_0, \ldots]$ rather than topological spaces using the standard equivalence relations.