Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
2  
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

1 Answer 1

In this article: http://arxiv.org/abs/1306.6063 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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.