Reducible forms, and forms of small slice rank

In this talk I will illustrate classical and recent results on secant varieties of the variety of reducible polynomials, focusing on the case of cubics. In particular, starting from classical determinantal equations detecting reducibility, I will show several generalizations to the setting of their secant varieties. The methods are based on the representation theory of certain shifted partial derivatives maps, the study of singularities of associated hypersurfaces, and on the classical geometry of defective varieties in small dimension. This is based on joint work with C. Flavi, A. Oneto and E. Ventura.