Defectivity and Identifiability: a degeneration approach to a generalized Waring Problem

The classical Waring problem for homogeneous polynomials can be translated into geometric terms, using the notion of defectivity and identifiability for secant varieties. The defectivity problem was completely solved by Alexander-Hirschowitz using classical degeneration techniques. On the other hand identifiability has recently been addressed by Mella and Galuppi. In this talk I will briefly explain the relationship between defectivity and identifiability in a more general setting and give bounds for a generalized Waring problem, introduced by Fröberg, Ottaviani and Shapiro. In particular we will see how the union of classical degeneration techniques combine with techniques borrowed from toric geometry, allowing us to give very sharp bounds on identifiability and defectivity in a much more general context. In the last part of the talk I will show how to generalize the previous approach to singular toric varieties. This is a joint work (in progress) with Elisa Postinghel.