Some examples of non-massive Frobenius manifolds in singularity theory

Let f,g:C2→Cf,g:C2→C be two quasi-homogeneous polynomials. We compute the VV-filtration of the restriction of ff to any plane curve Ct=g−1(t)Ct=g−1(t) and show that the Gorenstein generator dx∧dy/dgdx∧dy/dg is a primitive form. Using results of A. Douai and C. Sabbah, we conclude that the base space of the miniversal unfolding of ft≔f|Ctft≔f|Ct is a Frobenius manifold. At the singular fibre C0C0 we obtain a non-massive Frobenius manifold.