Jacobian Schemes of Conic-Line Arrangements and Eigenschemes

The Jacobian scheme of a reduced, singular projective plane curve is the zero- dimensional scheme, whose homogeneous ideal is generated by the partials of its defining poly- nomial. The degree of such a scheme is called the global Tjurina number and, if the curve is not a set of concurrent lines, some upper and lower bounds depending on the degree of the curve and the minimal degree of a Jacobian syzygy, have been given by A.A. du Plessis and C.T.C. Wall. In this paper we give a complete geometric characterization of conic-line arrangements, with global Tjurina number attaining the upper bound. Furthermore, we characterize conic-line arrangements attaining the lower bound for the global Tjurina number, among all curves with a linear Jacobian syzygy. As an application, we characterize conic-line arrangements with Jacobian scheme equal to an eigenscheme of some ternary tensor, and we study the geometry of their polar maps.