Topological Restrictions for $ k=2$      
A rational curve of degree $ d$ in $ {\mathbb{P}}^2$ with $ \kappa$ cusps has

$\displaystyle g\ :=\ (d-1)(d-2)/2 -\kappa \ $   nodes.$\displaystyle $

Types of nodes:     Real node         Isolated point         Conjugate pair

Theorem. (Kharlamov-S.)     $\displaystyle \char93 $ Real nodes$\displaystyle \ \leq\ (d-2)(d-3)/2 $
Proof uses Plücker and Klein formulae.