In this question, it was mentioned that an automorphism on a surface $X$ lifts to an automorphism on the blow up $p:\tilde X \to X$ at a point. I have trouble seeing this.
I know about the universal property of blow ups, but it does not seem to be sufficient.
Let $f: X \to X$ an automorphism with inverse $g$, which fixes the point we blow up. By the universal propery of blow-ups, we get morphisms $\bar f: X \to \tilde X$ and $\bar g: X \to \tilde X$
and thus morphisms $\tilde f = \bar f \circ p: \tilde X \to \tilde X$ and $\tilde g = \bar g \circ p: \tilde X \to \tilde X$.
Their composition $\tilde f \circ \tilde g: \tilde X \to \tilde X$ should be the identity, but we don't get this from the above. We only get $p = p \circ \tilde f\circ \tilde g$, so $\tilde f\circ \tilde g$ might swap points on the exceptional divisor.