Let $\mathcal{E}$ be a rank three vector bundle on $\mathbb{P}^1$. It splits as $\mathcal{E}\cong\mathcal{O}(a_1)\oplus\mathcal{O}(a_2)\oplus\mathcal{O}(a_3)$. It's not hard to see that any automorphism of $\mathbb{P}(\mathcal{E})$ acts trivially on the Picard group, which is generated by the class $f$ of a fiber and the class $h$ of $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$. In particular, because any automorphism takes fibers to fibers, we obtain a surjective map $\text{Aut}(\mathbb{P}(\mathcal{E}))\rightarrow \text{Aut}(\mathbb{P}^1)$. We get an exact sequence $$ 1\rightarrow G\rightarrow \text{Aut}(\mathbb{P}(\mathcal{E}))\rightarrow \text{Aut}(\mathbb{P}^1)\rightarrow 1 $$ We think of $G$ as automorphisms that fix each fiber as a set and leave $\mathbb{P}^1$ fixed. Algebraically, we can see $G=\mathbb{P}\text{Aut}(\mathcal{E})$. We have $\text{Aut}(\mathcal{E})\subset \text{End}(\mathcal{E})=H^0(\mathbb{P}^1,\mathcal{E}\otimes \mathcal{E}^{\vee})$. We can view an element of $H^0(\mathbb{P}^1,\mathcal{E}\otimes \mathcal{E}^{\vee})$ as a $3\times 3$ matrix with entries $a_{ij}\in H^0(\mathbb{P}^1,\mathcal{O}(a_i-a_j))$. I'd like to get my hands on what these automorphisms do in a geometric sense. It may be instructive to consider a specific example. Let $\mathcal{E}=\mathcal{O}(3)\oplus\mathcal{O}(3)\oplus\mathcal{O}(4)$. Our matrix is $\begin{pmatrix} a & b & 0 \\ c & d & 0 \\ l_1 & l_2 & e \end{pmatrix} $ where $a,b,c,d,e$ are scalars and $l_1, l_2\in H^0(\mathbb{P}^1,\mathcal{O}(1))$. The condition that this matrix defines an automorphism is that the determinant doesn't vanish, which just means that $(ad-bc)e\neq 0$. After projectivizing, we can set $e=1$. Intuitively, this matrix breaks up in to a few parts. The top left $2\times 2$ block seems to be a $GL_2$. The bottom left part is a $H^0(\mathcal{O}(1))^{\oplus 2}=\mathbb{A}^4$. How can I interpret these parts of the matrix geometrically? What types of automorphisms do they correspond to? One idea I have is the following. There is a natural divisor $\mathbb{P}(\mathcal{O}(3)\oplus\mathcal{O}(4))$ inside $\mathbb{P}(\mathcal{E})$ given by one of the obvious inclusions. Using the well-known Chow ring of $\mathbb{P}(\mathcal{E})$, I was able to compute that this divisor lies in class $h-3f$. Using the projection formula, we see that $h^0(\mathbb{P}(\mathcal{E}),\mathcal{O}(h-3f))=4$. From here, it seems like the bottom left part of the matrix moves around divisors in the class $h-3f$. Is this a reasonable interpretation? What about the $GL_2$ block? Perhaps it corresponds in some way to the divisor $\mathbb{P}(\mathcal{O}(3)\oplus\mathcal{O}(3))$?



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