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The plane Cremona Group $Cr_2(\mathbb{C})$ is the group of birational automorphisms of $\mathbb{P^2_{\mathbb{C}}}$. There are a lot of articles about this group (over $\mathbb{C}$). But I can't find anything about $Cr_2(\mathbb{R})$. What is known about this group? Why it's so difficult to study this group?

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More generally you can consider $Cr_2(K)$ for any subfield. Many of the results known for $Cr_2(C)$ give the corresponding results for $Cr_2(K)$ as particular cases. On the other hand, the generating and presentation issues are more specific, but in this case I think that there are some results of Iskovskikh in arbitrary perfect fields. – YCor Oct 29 '12 at 14:39
Over an algebraically closed field, life is easy. That is why people study things over $\mathbb{C}$ and not $\mathbb{R}$. There are some results over $\mathbb{R}$, like for example the study of birational maps which induce diffeomorphisms, but much less than in the case of the complex numbers. What kind of results do you need? – Jérémy Blanc Nov 1 '12 at 0:13

In this article: you can find some descriptions of the generators of the Cremona group over the reals, and get some other results. See also the references inside.

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