Let $X$ be a smooth and irreducible projective variety over $\mathbb{R}$ of dimension two. I am looking for an instance of such a variety where two distinct connected components of $X(\mathbb{R})$ are homeomorphic to the real projective plane $\mathbb{RP}^2$. Does someone know how to construct such a surface?


This is of course trivial if we don't want $X$ to be irreducible, since then we can just take the disjoint union of two projective planes.



Smooth, irreducible surface with real part containing two projective planes
[**Corrected**]


Any double cover of ${\bf RP}^2$ whose branch locus has 
degree $4n$ and no real component should do. An example is
the surface $y^2 = x\_0^4 + x\_1^4 + x\_2^4$ in the weighted projective space
whose coordinates $(x\_0:x\_1:x\_2::y)$ have degrees $1,1,1,2$.
Thus $(x\_0:x\_1:x\_2::y)$ is equivalent to
$(\lambda x\_0:\lambda x\_1:\lambda x\_2::\lambda^2 y)$ 
for nonzero $\lambda \in \bf R$, so there are well-defined components
$y>0$ and $y<0$, and the projection to $(x\_0:x\_1:x\_2)$
maps each component homeomorphically to ${\bf RP}^2$.


[Earlier I rashly omitted the "degree $4n$" condition and proposed
the two-sheeted hyperboloid $y^2 = x\_0^2 + x\_1^2 + x\_2^2$ as an example.
That's clearly wrong because it's isomorphic (as a surface in ${\bf RP}^3$)
with a Euclidean sphere.]


