Let L : dom L ⊂ L2(Ω) → L2(Ω) be a self-adjoint operator, Ω being open and bounded in RN. We give a description of the Fučík spectrum of L away from the essential spectrum. Let λ be a point in the discrete spectrum of L; provided that some non-degeneracy conditions are satisfied, we prove that the Fučík spectrum consists locally of a finite number of curves crossing at (λ, λ). Each of these curves can be associated to a critical point of the function H : x ↦ 〈|x|,x〉L2 restricted to the unit sphere in ker(L – λI). The corresponding critical values determine the slopes of these curves. We also give global results describing the Fučík spectrum, and existence results for semilinear equations, by performing degree computations between the Fučík curves.