# Automorphism groups of Hirzebruch surfaces

The Hirzebruch surfaces are the $\mathbb{P}^1$ bundles $\mathbb{F_n}$ ($n\geqslant 0$) which can be obtained projectivizing the rank $2$ vector bundles $\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(n)$. Those surfaces are famous because they exhaust ($n\neq 1$) the minimal rational (smooth, projective) surfaces other than $\mathbb{P}^2$. The most familiar examples are $\mathbb{F_0}\simeq \mathbb{P}^1\times\mathbb{P}^1$ and $\mathbb{F_1}$ which is isomorphic to $\mathbb{P}^2$ blown up at a point. Other features of Hirzebruch surfaces are apparently well-known, such as its intersection theory.

My question is about automorphism groups of Hirzebruch surfaces. For the sake of comparation, the automorphism group of $\mathbb{P}^2$ is isomorphic to $PGL(3,\mathbb{C})$ and I manage to find a very nice description of $Aut(\mathbb{F_0})$ in this answer (Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$), since a quadric smooth projective surface $Q\subset \mathbb{P}^3$ is isomorphic to $\mathbb{P}^1\times\mathbb{P}^1$.

Do we have a similar description of $Aut(\mathbb{F_n})$ for positive $n$?

• you can see page 12 of mimuw.edu.pl/~jarekw/EAGER/pdf/FiniteSubgroupsCremona.pdf – Chen Jiang Mar 17 '18 at 3:38
• Namely it's isomorphic to a semidirect product $V_n\rtimes (\mathrm{GL}_2/\mu_n)$ where $V_n$ is the $n$-th symmetric power of the standard 2-dimensional rep (so $\dim(V_n)=n+1$) and $\mu_n$ is the group of $n$-th roots of unity in the group of scalar matrices. – YCor Mar 17 '18 at 7:08

For $n \ge 2$ the surface $\mathbb{F}_n$ is the blowup of the weighted projective plane $\mathbb{P}(1,1,n)$ at its singular point. Because of that $$Aut(\mathbb{F}_n) \cong Aut(\mathbb{P}(1,1,n)).$$ The automorphism group of $\mathbb{P}(w_1,w_2,\dots,w_m)$ is the group of non-degenerate $m$-by-$m$ matrices with the $(i,j)$-entry being a polynomial of degree $w_i - w_j$, up to simultaneous rescaling.