Let $C$ be a smooth projective curve. Is it true that


$$\textrm{Aut}(C\times C)\cong S\_2 \ltimes (\textrm{Aut}(C)\times \textrm{Aut}(C))$$


and in case, what would be a reference for this? Thanks.



Automorphisms of cartesian products of curves
This is a particular case of a more general rigidity result, whose proof (similar to the one given in abx's answer) can be found in Lemma 3.8 of 


F. Catanese, [*Fibred surfaces, varieties isogenous to a product and related moduli spaces*](http://muse.jhu.edu/journals/american_journal_of_mathematics/v122/122.1catanese.pdf), *Amer. J. Math.* **122** (2000), no. 1, 1-44.



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> **Proposition.** Let $f \colon C\_1 \times C\_2 \longrightarrow B\_1 \times B\_2$ be a surjective holomorphic map between products of curves. Assume that both $B\_1$ and $B\_2$ have genus $\geq 2$. Then, after possibly exchanging $B\_1$ with $B\_2$, there are holomorphic maps $f\_i \colon C\_i \to B\_i$ such that $$f(x, \, y) = (f\_1(x), \, f\_2(y)).$$
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