Classifying real polynomial pencils

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