Given a bounded smooth domain Ω ⊂ Rn, n ≥ 2, and a second-order elliptic self-adjoint operator A on Ω, the set of points (α, β) ∈ R2 for which the problem Au = αu+ − βu- in Ω, u = 0 on ∂Ω (where u± = max{±u, 0}), has a non-trivial solution is called the Fucik spectrum of A. In this note we extend some recent results of Pistoia on the structure of this set for generic operators A (the genericity is with respect to the domain Ω or the coefficients of A).