Intersections of connected components of real curves

Question

Let $C_1,C_2\subset \mathbb{P}^2(\mathbb{R})$ be real algebraic curves each of degree $d$. By Bezout's Theorem these curves have at most $d^2$ points of intercestion. Since we are in the real case, each of the two curves may be a union of connected components. (By Harnack's theorem each of the curve may be decomposed into $\frac{(d-1)(d-2)}{2}+1$ may connected components.) Now I wonder, what can be said about the number of intersections of the various connected components. For example, suppose that $C_1$ has only one connected component but $C_2$ has two of them. Is it then possible that $C_1$ has $d^2$ many intersections wit onlyh one of the connected components of $C_2$?

Answer 1

This makes me think of Harnack's construction (cf. Bochnak, Coste, Roy: Real algebraic geometry, Proposition 11.6.3): $C_2$ is an $M$-curve of any degree $d$, i.e., having the maximal number of connected components in degree $d$, such that there is a real projective line $L$ intersecting one connected component of $C_2$ in $d$ real points. Now take generic real projective lines $L_1,\ldots,L_d$ sufficiently close to $L$ and let $C_1$ be a suitable perturbation of their union. Then $C_1$ intersects $C_2$ in $d^2$ real points, all of them on one connected component of $C_2$.