It has been shown recently that, under some generic assumptions, there exists a Hopf curve lambda = lambda (epsilon) for singularly perturbed systems of the form (x) over dot = f (x, y, lambda), epsilon (y) over dot = g(x, y, lambda) near the singular surface defined by det g(y) = 0. In this note, we are concerned With the Hopf curve and obtain three results. 1) We prove that the eigenvalue crossing condition for the Hopf curve holds without additional assumption. 2) We provide an improved form of an existing derivative formula for the Hopf curve which is more suitable for practical computations. 3) We give a quite precise description of the spectrum structure of the linearization along the Hopf curve. All three results (stated in the main theorem) are useful for a better understanding of Hopf bifurcations in singularly perturbed systems. Our analysis is based on a factorization of parameter dependent polynomials (Lemma 2.3).