Geometry of weak-bitangent lines to quartic curves and sections on certain rational elliptic surfaces

It is well known that a smooth quartic curve has twenty-eight bitangent lines. For a reduced, possibly singular quartic curve, we introduce the notion of weak-bitangent line. This can be considered as a generalization of bitangent lines. In this article, we consider weak-bitangent lines for certain reduced quartic curves from the viewpoint of rational elliptic surfaces. We utilize Mumford representations of semi-reduced divisors in order to deal with equations of weak-bitangent lines for certain reduced quartic curves. As a result, we can give new proofs for some classical results on singular quartic curves and their bitangent lines.