We classify the subpencils of complete linear systems for the hyperplane sections on K3 surfaces obtained as the complete intersection of a hyperquadric and a hypercubic. The classification is done from three points of view, namely, the type of a general fibre, the base locus and the Horikawa index of the essential member. This classification shows the distinct phenomenons depending on the rank of the hyperquadrics containing the surface.

We will analyze the relationships between the special fibres of a pencil Λ of plane curve singularities and the Jacobian curve J of Λ (defined by the zero locus of the Jacobian determinant for any fixed basis ϕϕ′ ∈ Λ). From the results, we find decompositions of J (and of any special fibre of the pencil) in terms of the minimal resolution of Λ. Using these decompositions and the topological type of any generic pair of curves of Λ, we obtain some topological information about J. More precise decompositions for J can be deduced from the minimal embedded resolution of any pair of fibres (not necessarily generic) or from the minimal embedded resolution of all the special fibres.