(This is a part of Seminars on Algebraic Surfaces and Related Topics.)
In this talk, I will explain how to associate a nodal surface in P3 with a cubic hypersurface, generalizing the method by Togliatti who constructed quintics with 31 nodes via a discriminant of a nodal cubic 4-folds. For low degrees(≤5), these constructions help to understand the classification problem of nodal surfaces, especially when the surface has the maximal number of nodes. For higher degrees the things get more complicated. I will explain our recent result on sextics proving that every nodal sextics with maximal number of nodes admit Togliatti type descriptions. This talk is based on joint works with Fabrizio Catanese, Stephen Coughlan, Davide Frapporti, Michael Kiermaier, and Sascha Kurz.