Stability of Halphen pencils of index two

In this talk I will present some results about the GIT stability of Halphen pencils of index two under the action of SL(3). These are pencils of plane curves of degree six having nine (possibly infinitely near) base points of multiplicity two. Inspired by the work of Miranda on pencils of plane cubics, I will explain how we can explore the geometry of the associated rational elliptic surfaces.