Part of Hilbert's 16th problem is:
It seems to me that a thorough investigation of the relative positions of the upper bound for separate branches is of great interest, and similarly the corresponding investigation of the number, shape and position of the sheets of an algebraic surface in space (...)
The solution for a quartic surface is contained in Kharlamov: The topological type of nonsingular surfaces in RP3 of degree four.
What is the status of this problem? In particular, is there a conjectural upper bound for the maximum number of connected components of a smooth degree $d$ surface in $\mathbb{RP}^3$?
