Can the Hirzebruch Surface $F\_2:=\mathbb{P}(\mathcal{O}\oplus \mathcal{O}(2))$ be obtained by some GIT quotient of $\mathbb{P}^4$ (or $\mathbb{C}^4$)?



Hirzebruch Surface F2
Depending on the interpretation of your question, the answer is *Yes*.


In fact the Hirzebruch surface $\mathbb{F}\_n =\mathbb{P}(\mathcal{O}\oplus \mathcal{O}(n))$ is the quotient of $X = \mathbb{A}^2 \setminus \{0\} \times \mathbb{A}^2 \setminus \{0\}$ with respect to the group action
$$ \mathbb{G}\_m^2 \times X \to X, \quad (\lambda,\mu) \cdot (s,t;x,y) = (\lambda s, \lambda t; \mu x, \lambda^{-n} \mu y).$$


From a highbrow perspective, such a realisation exists as $\mathbb{F}\_n$ is toric. (Every toric variety is a quotient of an open subset of an affine space by the action of some multiplicative group, by the theory of Cox rings).


