I want to play around with real-algebraic geometry, but am not sure where to start. I have a couple basic questions which motivate my study:
- How can I come up with explicit models for smooth surfaces
- How can I extract topological information to figure out which diffeomorphism class they live in
Since this is determined by the euler characteristic and orientability, does there exist algebraic tools to determine this for a smooth complete intersection variety in $\mathbb{RP}^n$?
1) Explicit models for (spacial) surfaces comes often from perturbing surfaces of smaller degree. As an example, surface of degree $1$ are exactly plane $ H \subset \Bbb RP^3$. For degree two you can have a sphere or a torus. For degree $3$, taking the union of one plane and one sphere/torus and perturbing the reunion suitably can gives you all the topological types, which is if I remember well a connected sum of a bunch of $\Bbb RP^2$, you can have $1,3,5$ or $7$. Another way to construct surfaces is to use patchworking. As constructing interesting surfaces is already not trivial, I don't know how you can ensure that they are complete intersection. And even if you understand well the topology of some hypersurfaces, computing the topological type of the intersection seems pretty hard to me.
2) For extracting topological information, there is the Thom-Smith inequality and the Comessati inequalities (and probably other I am not aware of) which can gives you bound on the Euler characteristic/Betti numbers of your real surfaces in function of the Hodge numbers of the associated complex surface. If you know how to compute them you will have several restrictions at least.