Suppose $X$ and $Y$ are smooth real algebraic surfaces in $\mathbb P^3(\mathbb R)$. If X and Y are birational over the reals, then is it true that they also have the same number of connected components (with the topology of $\mathbb P^3(\mathbb R)$ as a real manifold)? Is there an example of birational real surfaces in 3-space that are not diffeomorphic?
2026-03-25 06:02:00.1774418520
connected components of birational real surfaces
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I now know the answer. This can be found in Real Algebraic Geometry by Bochnak, Coste and Roy. Theorem 3.4.12 says that the number of connected components is a birational invariant.