Let $X$ be a projective surface and let $x\in X$ be a smooth point. Consider the blow up $Bl\_{x}X$ of $X$ in $x$, and let $E$ be the exceptional divisor. Suppose we know that $E$ is the only (-1)-curve in $Bl\_{x}X$, so that $\alpha(E) = E$ for any $\alpha\in Aut(Bl\_{x}X)$. Then any automorphisms $\alpha\in Aut(Bl\_{x}X)$ induces an automorphism of $X$ fixing $x$, let us denote $Aut\_{x}(X)\subseteq Aut(X)$ the subgrup of this automorphisms. We get a injective morphism
$$f:Aut(Bl\_{x}X)\rightarrow Aut\_{x}(X).$$
What can one say about the image of this morphism? Are there cases where $f$ is an isomorphism, or on the contrary cases in which the image is trivial? In the simple case $X = \mathbb{P}^{2}$, what happens?



Lifting automorphisms on a Blow-up surface
Let $\sigma:X\to X$ be an automorphism fixing a point $x\in X$. Then since the inverse image of $x$ is a Cartier divisor in both $Bl\_x X$ and $Bl\_x \sigma(X)$, by the universal property of blow-ups, we get an automorphism $\tilde{\sigma}$ of the blow-up $Bl\_x X$. Since $\tilde{\sigma}$ restricts to $\sigma$ on the complement of the exceptional divisor, it is clear that $\sigma$ is in the image of $f$.


