Classifying real polynomial pencils

Let ${\text{bP}}^n$ be the space of all homogeneous polynomials of degree $n$ in two variables with real coefficients. The standard discriminant $\D_{n+1}\subset \bP^n$ is Whitney stratified according to the number and the multiplicities of multiple real zeros. A real polynomial pencil, that is, a line $L\subset {\text{bP}}^n$ is called generic if it intersects ${\text{D}}_{n+1}$ transversally. Nongeneric pencils form the Grassmann discriminant ${\text{D}}_{2,n+1}\subset \text{gtn}$, where $\text{gtn}$ is the Grassmannian of lines in ${\text{bP}}^n$. We enumerate the connected components of the set $\widetilde{\text{gtn}}=\text{gtn}\setminus {\text{D}}_{2,n+1}$ of all generic lines in ${\text{bP}}^n$ and relate this topic to the Hawaii conjecture and the classical theorems of Obreschkoff and Hermite-Biehler.