Stability of pencils of cubic surfaces in P3

In this paper we study the classification of pencils of cubic surfaces in P-3, up to projective equivalence. We obtain explicit vanishing criteria on the Plucker coordinates of a pencil for both stability and semi-stability; moreover, we give the equations defining pairs of generators for unstable and not properly stable pencils. Thus we extend the work of Miranda and Ballico [5, 1]. We give some geometric criteria for when a pencil is properly stable, and in particular, we give a characterization of smooth not properly stable pencils.